chapter 1 equations, inequalities, and mathematical modelingsection 1.1 graphs of equations 69 4....

126
CHAPTER 1 Equations, Inequalities, and Mathematical Modeling Section 1.1 Graphs of Equations . . . . . . . . . . . . . . . . . . . . 68 Section 1.2 Linear Equations in One Variable . . . . . . . . . . . . . 76 Section 1.3 Modeling with Linear Equations . . . . . . . . . . . . . . 90 Section 1.4 Quadratic Equations and Applications . . . . . . . . . . 104 Section 1.5 Complex Numbers . . . . . . . . . . . . . . . . . . . . 121 Section 1.6 Other Types of Equations . . . . . . . . . . . . . . . . . 128 Section 1.7 Linear Inequalities in One Variable . . . . . . . . . . . . 147 Section 1.8 Other Types of Inequalities . . . . . . . . . . . . . . . . 157 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

Upload: others

Post on 06-Aug-2020

0 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

C H A P T E R 1Equations, Inequalities, and Mathematical Modeling

Section 1.1 Graphs of Equations . . . . . . . . . . . . . . . . . . . . 68

Section 1.2 Linear Equations in One Variable . . . . . . . . . . . . . 76

Section 1.3 Modeling with Linear Equations . . . . . . . . . . . . . . 90

Section 1.4 Quadratic Equations and Applications . . . . . . . . . . 104

Section 1.5 Complex Numbers . . . . . . . . . . . . . . . . . . . . 121

Section 1.6 Other Types of Equations . . . . . . . . . . . . . . . . . 128

Section 1.7 Linear Inequalities in One Variable . . . . . . . . . . . . 147

Section 1.8 Other Types of Inequalities . . . . . . . . . . . . . . . . 157

Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

Page 2: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

C H A P T E R 1Equations, Inequalities, and Mathematical Modeling

Section 1.1 Graphs of Equations

68

■ You should be able to use the point-plotting method of graphing.

■ You should be able to find x- and y-intercepts.

(a) To find the x-intercepts, let and solve for x.

(b) To find the y-intercepts, let and solve for y.

■ You should be able to test for symmetry.

(a) To test for x-axis symmetry, replace y with

(b) To test for y-axis symmetry, replace x with

(c) To test for origin symmetry, replace x with and y with

■ You should know the standard equation of a circle with center and radius r:

�x � h�2 � �y � k�2 � r 2

�h, k�

�y.�x

�x.

�y.

x � 0

y � 0

1.

(a)

Yes, the point is on the graph.

2 � 2

�0, 2�: 2 �? �0 � 4

y � �x � 4

(b)

Yes, the point is on the graph.

3 � �9

�5, 3�: 3 �? �5 � 4

2.

(a)

Yes, the point is on the graph.

0 � 0

4 � 6 � 2 �?

0

�2, 0�: �2�2 � 3�2� � 2 �?

0

y � x2 � 3x � 2

(b)

No, the point is not on the graph.

12 � 8

4 � 6 � 2 �?

8

��2, 8�: ��2�2 � 3��2� � 2 �?

8

3.

(a)

No, the point is not on the graph.

5 � 4 � 1

�1, 5�: 5 �? 4 � �1 � 2�

y � 4 � �x � 2�(b)

Yes, the point is on the graph.

0 � 4 � 4

�6, 0�: 0 �? 4 � �6 � 2�

Vocabulary Check

1. solution or solution point 2. graph 3. intercepts

4. axis 5. circle; 6. point-plotting�h, k�; ry-

Page 3: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.1 Graphs of Equations 69

4.

(a)

Yes, the point is on the graph.

�163 � �16

3

83 �243 �

?�16

3

83 � 8 �?

�163

13 � 8 � 2 � 4 �?

�163

�2, �163 �: 13�2�3 � 2�2�2 �

?�16

3

y �13x3 � 2x2

(b)

No, the point is not on the graph.

�27 � 9

�9 � 18 �?

9

13��27� � 2�9� �?

9

��3, 9�: 13��3�3 � 2��3�2 �?

9

5.

x541−1−2−3

4

7

2

1

−1

2

3

5

y

y � �2x � 5

7.

1 2

3

4

5

4 5x

−2 −1−1

−2

yy � x2 � 3x

x 0 1 2

y 7 5 3 1 0

�52, 0��2, 1��1, 3��0, 5���1, 7��x, y�

52�1

x 0 1 2 3

y 4 0 0

�3, 0��2, �2��1, �2��0, 0���1, 4��x, y�

�2�2

�1

6. y �34x � 1

0 1 2

0 12�

14�1�

52

43�2

8. 5 � x2

�2 �1 0 1 2

1 4 5 4 1

–4 –3 –2 –1 2 3 4

–4

–3

–2

1

2

3

4

x

y

–4 –3 –1 1 3 4

–2

–1

1

2

3

6

4

x

y 9.

x-intercepts:

y-intercept:

�0, 16�

y � 16 � 4�0�2 � 16

��2, 0�, �2, 0�

x � ±2

x2 � 4

4x2 � 16

0 � 16 � 4x2

y � 16 � 4x2

Page 4: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

70 Chapter 1 Equations, Inequalities, and Mathematical Modeling

10.

�0, 9�

y � 9

y � 32

y-intercept: y � �0 � 3�2

(�3, 0)

x � �3

0 � x � 3

x-intercept: 0 � �x � 3�2

y � �x � 3�2 11.

x-intercepts:

or

y-intercept:

�0, 0�

y � 0

y � 2�0�3 � 4�0�2

�0, 0�, �2, 0�

x � 2 x � 0

0 � 2x2�x � 2�

0 � 2x3 � 4x2

y � 2x3 � 4x2

12.

�0, 1�, �0, �1�

y � ±1

y-intercepts: y2 � 0 � 1

��1, 0�

x � �1

x-intercept: 0 � x � 1

y2 � x � 1 13. y-axis symmetry

–4 –3 –1 1 3 4

1

2

−2

3

4

x

y

14.

1 2 3 6 7 8

−4

−3

−2

−1

1

2

3

4

x

y 15. Origin symmetry

–4 –3 –2 1 2 3 4

–4

–3

–2

1

2

3

4

x

y

16.

–4 –3 –2 –1 1 2 3 4

–4

–3

–2

1

2

3

4

x

y

17.

y-axis symmetry

No x-axis symmetry

No origin symmetry ��x�2 � ��y� � 0 ⇒ x2 � y � 0 ⇒

x2 � ��y� � 0 ⇒ x2 � y � 0 ⇒

��x�2 � y � 0 ⇒ x2 � y � 0 ⇒

x2 � y � 0 18.

x-axis symmetry

x � y2 � 0

x � ��y�2 � 0

x � y2 � 0

19.

No y-axis symmetry

No x-axis symmetry

Origin symmetry�y � ��x�3 ⇒ �y � �x3 ⇒ y � x3 ⇒

�y � x3 ⇒ y � �x3 ⇒

y � ��x�3 ⇒ y � �x3 ⇒

y � x3 20.

y-axis symmetry

y � x4 � x2 � 3

y � ��x�4 � ��x�2 � 3

y � x4 � x2 � 3

Page 5: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.1 Graphs of Equations 71

21.

No y-axis symmetry

No x-axis symmetry

Origin symmetry�y ��x

��x�2 � 1 ⇒ �y �

�xx2 � 1

⇒ y �x

x2 � 1 ⇒

�y �x

x2 � 1 ⇒ y �

�xx2 � 1

y ��x

��x�2 � 1 ⇒ y �

�xx2 � 1

y �x

x2 � 1

22.

y-axis symmetry

y � �9 � x2

y � �9 � ��x�2

y � �9 � x2

23.

No y-axis symmetry

x-axis symmetry

No origin symmetry��x���y�2 � 10 � 0 ⇒ �xy2 � 10 � 0 ⇒

x��y�2 � 10 � 0 ⇒ xy2 � 10 � 0 ⇒

��x�y2 � 10 � 0 ⇒ �xy2 � 10 � 0 ⇒

xy2 � 10 � 0

24.

Origin symmetry

xy � 4

��x���y� � 4

xy � 4 25.

x-intercept:

y-intercept:

No axis or origin symmetry

�0, 1�

�13, 0�

( (x

(0, 1) 13

y

−1−2−3−4 1 2 3 4−1

−2

−3

1

4

5

, 0

y � �3x � 1

26.

x-intercept:

y-intercept:

No symmetry

�0, �3�

�32, 0�

(0, −3)

32 , 0( )

y

x−1−2−3 1 2 3

−1

−2

−3

1

2

y � 2x � 3 27.

Intercepts:

No axis or origin symmetry

�0, 0�, �2, 0�

−1

2

3

4

−1 1 2 3 4x

−2

−2

(0, 0) (2, 0)

yy � x2 � 2x

x 0 1 2 3

y 3 0 0 3�1

�1

28.

x-intercept:

y-intercept:

No symmetry

�0, 0�

��2, 0�, �0, 0�x

321−1−3−4−5

−6

−5

−4

−3

−2

2

−1

(−2, 0) (0, 0)

yy � �x2 � 2x 29.

Intercepts:

No axis or origin symmetry

−4 −3 −2−1

1

2

4

5

6

7

1 2 3 4x

(0, 3)

−3, 03( (

y�0, 3�, � 3��3, 0�y � x3 � 3

x 0 1 2

y 2 3 4 11�5

�1�2

Page 6: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

72 Chapter 1 Equations, Inequalities, and Mathematical Modeling

30.

x-intercept:

y-intercept:

No symmetry

�0, �1�

�1, 0�

–3 –2 –1 2 3

–4

–3

–2

1

2

x(1, 0)

(0, −1)

yy � x3 � 1 31.

Domain:

Intercept:

No axis or originsymmetry

�3, 0��3, ��

1 2 3 4 5 6–1

1

2

3

4

5

x(3, 0)

yy � �x � 3

x 3 4 7 12

y 0 1 2 3

32.

Domain:

y-intercept:

No symmetry

�0, 1�

���, 1�

–4 –3 –2 –1 1 2–1

2

3

4

5

x(1, 0)

(0, 1)

yy � �1 � x

33.

Intercepts:

No axis or origin symmetry

�0, 6�, �6, 0�

2−2

2

−2

4

6

8

10

12

4 6 8 10 12x

(6, 0)

(0, 6)

yy � �x � 6�

x 0 2 4 6 8 10

y 8 6 4 2 0 2 4

�2

34.

x-intercepts:

y-intercept:

y-axis symmetry

�0, 1�

�±1, 0�

x2 31−2−3

2

−1

−2

−3

1

3

(−1, 0) (1, 0)

(0, 1)

yy � 1 � �x� 35.

Intercepts:

x-axis symmetry

–2 1 2 3 4

–3

–2

2

3

x

(0, −1)

(−1, 0) (0, 1)

y

�0, �1�, �0, 1�, ��1, 0�

x � y2 � 1

x 0 3

y 0 ±2±1

�1

36.

x-intercept:

y-intercept:

x-axis symmetry

�0, ±�5 ���5, 0�

x21−1−2−3−4−6

4

−1

−4

−3

1

3

(−5, 0)

0, 5( )

0, − 5( )

yx � y2 � 5 37.

Intercepts: �6, 0�, �0, 3�

−10

−10

10

10

y � 3 �12x

Page 7: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.1 Graphs of Equations 73

38.

Intercepts: �0, �1�, �32, 0�

−10

−10

10

10

y �23x � 1 39.

Intercepts: �3, 0�, �1, 0�, �0, 3�

−10 10

−10

10

y � x2 � 4x � 3 40.

Intercepts: ��2, 0�, �1, 0�, �0, �2�

−10

−10

10

10

y � x2 � x � 2

41.

Intercept: �0, 0�

−10

−10

10

10

y �2x

x � 142.

Intercept: �0, 4�

−10

−10

10

10

y �4

x2 � 143.

Intercept: �0, 0�

−10

−10

10

10

y � 3�x

44.

Intercepts: ��1, 0�, �0, 1�

−10

−10

10

10

y � 3�x � 1 45.

Intercepts: �0, 0�, ��6, 0�

−10

−10

10

10

y � x�x � 6 46.

Intercepts: �0, 0�, �6, 0�

−10

−10

10

10

y � �6 � x��x

47.

Intercepts: ��3, 0�, �0, 3�

−10

−10

10

10

y � �x � 3� 48.

Intercepts: �±2, 0�, �0, 2�

−10

−10

10

10

y � 2 � �x� 49. Center: radius: 4

Standard form:

x2 � y2 � 16

�x � 0�2 � �y � 0�2 � 42

�0, 0�;

53. Center: solution point:

Standard form: �x � 1�2 � �y � 2�2 � 5

�0 � 1�2 � �0 � 2�2 � r 2 ⇒ 5 � r2

�x � ��1��2 � �y � 2�2 � r2

�0, 0���1, 2�; 54.

�x � 3�2 � �y � 2�2 � 25

�x � 3�2 � �y � ��2��2 � 52

� �42 � ��3�2 � �25 � 5

r � ��3 � ��1��2 � ��2 � 1�2

50.

x2 � y2 � 25

�x � 0�2 � �y � 0�2 � 52 51. Center: radius: 4

Standard form:

�x � 2�2 � �y � 1�2 � 16

�x � 2�2 � �y � ��1��2 � 42

�2, �1�; 52.

�x � 7�2 � �y � 4�2 � 49

�x � ��7��2 � �y � ��4��2 � 72

Page 8: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

74 Chapter 1 Equations, Inequalities, and Mathematical Modeling

55. Endpoints of a diameter:

Center:

Standard form: �x � 3�2 � �y � 4�2 � 25

�0 � 3�2 � �0 � 4�2 � r 2 ⇒ 25 � r 2

�x � 3�2 � �y � 4�2 � r2

�0 � 62

, 0 � 8

2 � � �3, 4�

�0, 0�, �6, 8� 56.

Midpoint of diameter (center of circle):

x2 � y2 � 17

�x � 0�2 � �y � 0�2 � ��17�2

��4 � 4

2,

�1 � 1

2 � � �0, 0�

�1

2�68 � �1

2��2��17 � �17

�1

2�64 � 4

�1

2���8�2 � ��2�2

r �1

2���4 � 4�2 � ��1 � 1�2

57.

Center: radius: 5

1−1−2

−2−3−4

−6

−3−4

1234

6

2 3 4 6x

y

(0, 0)

�0, 0�,

x2 � y2 � 25 58.

Center: radius: 4

x

−2

−3

1

2

3

5

−5

531−1−2−3−5 2

y

(0, 0)

�0, 0�,

x2 � y2 � 16 59.

Center: radius: 3

−3 −2 1

1

2 4 5x

−1

−2

−3

−4

−5

−6

−7

y

(1, −3)

�1, �3�,

�x � 1�2 � �y � 3�2 � 9

60.

Center: radius: 1

–2 –1 1 2

–1

1

3

x

y

(0, 1)

�0, 1�,

x2 � �y � 1�2 � 1 61.

Center: radius:

–1 1 2 3

1

3

x

y

( )12

, 12

32�1

2, 12�,�x �

12�2

� �y �12�2

�94 62.

Center: radius:

y

x−1 1 2 3 4

−1

−2

−3

−4

1

(2, −3)

43�2, �3�,

�x � 2�2 � �y � 3�2 �169

63.

250,000

200,000

150,000

100,000

50,000

1 2 3 4 5 6 7 8t

Year

Dep

reci

ated

val

ue

y

y � 225,000 � 20,000t, 0 ≤ t ≤ 8 64.

t1 65432

1600

3200

4800

6400

8000

Year

Dep

reci

ated

val

ue

y

y � 8100 � 929t, 0 ≤ t ≤ 6

Page 9: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.1 Graphs of Equations 75

65. (a)

(c)

0 1800

8000

y

x

(b)

(d) When yards, the area is a maximum of square yards.

(e) A regulation NFL playing field is 120 yards long and yards wide. The actual area is 6400 square yards.531

3

751119x � y � 862

3

A � xy � x�5203 � x�

y �5203 � x

2y �1040

3 � 2x

2x � 2y �1040

3

66. (a)

(c)

0 1800

9000

y

x

(b) meters so:

(d) and

A square will give the maximum area of 8100 square meters.

(e) The dimensions of a Major League Soccer field can varybetween 110 and 120 yards in length and between 70 and 80yards in width.

y � 90x � 90

A � lw � x�180 � x�

w � y � 180 � x

2x � 2y � 360

P � 360

67.

(a) and (b)

Year (20 ↔ 1920)

Lif

e ex

pect

ancy

20 40 60 80 100

100

80

60

40

20

t

y

y � �0.0025t 2 � 0.574t � 44.25, 20 ≤ t ≤ 100

(c) For the year 1948, let years.

(d) For the year 2005, let years.

For the year 2010, let years.

(e) No. The graph reaches a maximum of yearswhen or during the year 2014. After this time,the model has life expectancy decreasing, which is notrealistic.

t 114.8,y 77.2

t � 110: y 77.1

t � 105: y 77.0

t � 48: y 66.0

68. (a)5 10 20 30 40 50 60 70 80 90 100

430.43 107.33 26.56 11.60 6.36 3.94 2.62 1.83 1.31 0.96 0.71y

x

(b) y

x20 40 60 80 100

50100150200250300350400450

Res

ista

nce

(in

ohm

s)

Diameter of wire (in mils)

(c) When

(d) As the diameter of the wire increases, the resistancedecreases.

y �10,77085.52 � 0.37 � 1.10327.

x � 85.5,

69. False. A graph is symmetric with respect to the axis if,whenever is on the graph, is also on thegraph.

�x, �y��x, y�x- 70. True. The graph can have no intercepts, one, two or

many. For example, a circle centered at the origin has two intercepts. A circle of radius 1, centered at has no intercepts.y-

�7, 7�,y-

Page 10: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

76 Chapter 1 Equations, Inequalities, and Mathematical Modeling

71. The viewing window is incorrect. Change the viewingwindow. Examples will vary. For example,will not appear in the standard window setting.

y � x2 � 2072.

(a)

To be symmetric with respect to the axis, can beany non-zero real number; must be zero.

(b)

To be symmetric with respect to the origin, must bezero; can be any non-zero real number.b

a

y � �ax2 � bx3

�y � ax2 � bx3

�y � a��x�2 � b��x�3

bay-

y � a��x�2 � b��x�3 � ax2 � bx3

y � ax2 � bx3

73.

Terms: 9x5, 4x3, �7

9x5 � 4x3 � 7 74. ��7 � 7 � 7 � 7� � ��7�4 � �74

75. �18x � �2x � 3�2x � �2x � 2�2x 76.

� 5��20 � 3� � 5�2�5 � 3�

�55��20 � 3�

20 � 9�

55��20 � 3�11

55

�20 � 3�

55�20 � 3

��20 � 3�20 � 3

77. 6�t2 � t2�6 � �t�1�3 � 3��t� 78. 3��y � � y1�2�1�3� y1�6 � 6�y

Section 1.2 Linear Equations in One Variable

■ You should know how to solve linear equations.

■ An identity is an equation whose solution consists of every real number in its domain.

■ To solve an equation you can:

(a) Add or subtract the same quantity from both sides.

(b) Multiply or divide both sides by the same nonzero quantity.

(c) Remove all symbols of grouping and all fractions.

(d) Combine like terms.

(e) Interchange the two sides.

■ Check the answer!

■ A “solution” that does not satisfy the original equation is called an extraneous solution.

■ Be able to find intercepts algebraically.

a � 0ax � b � 0,

Vocabulary Check

1. equation 2. solve 3. identities; conditional

4. 5. extraneousax � b � 0

Page 11: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.2 Linear Equations in One Variable 77

1.

(a)

is not a solution.

(c)

is a solution.x � 4

17 � 17

5�4� � 3 �?

3�4� � 5

x � 0

�3 � 5

5�0� � 3 �?

3�0� � 5

5x � 3 � 3x � 5

(b)

is not a solution.

(d)

is not a solution.x � 10

47 � 35

5�10� � 3 �?

3�10� � 5

x � �5

�28 � �10

5��5� � 3 �?

3��5� � 5

2.

(a)

is not a solution.

(c)

is not a solution.x � 8

�17 � 23

7 � 24 �?

40 � 17

7 � 3�8� �?

5�8� � 17

x � 8

x � �3

16 � �32

7 � 9 �?

�15 � 17

7 � 3��3� �?

5��3� � 17

x � �3

7 � 3x � 5x � 17

(b)

is not a solution.

(d)

is a solution.x � 3

�2 � �2

7 � 9 �?

15 � 17

7 � 3�3� �?

5�3� � 17

x � 3

x � 0

7 � �17

7 � 0 �?

0 � 17

7 � 3�0� �?

5�0� � 17

x � 0

3.

(a)

is a solution.

(c)

is not a solution.x � 4

51 � 30

48 � 8 � 5 �?

32 � 2

3�4�2 � 2�4� � 5 �?

2�4�2 � 2

x � �3

16 � 16

27 � 6 � 5 �?

18 � 2

3��3�2 � 2��3� � 5 �?

2��3�2 � 2

3x2 � 2x � 5 � 2x2 � 2

(b)

is a solution.

(d)

is not a solution.x � �5

60 � 48

75 � 10 � 5 �?

50 � 2

3��5�2 � 2��5� � 5 �?

2��5�2 � 2

x � 1

0 � 0

3 � 2 � 5 �?

2 � 2

3�1�2 � 2�1� � 5 �?

2�1�2 � 2

4.

(a)

is not a solution.

(c)

is not a solution.x � 0

�3 � �11

0 � 0 � 3 �?

0 � 0 � 11

5�0�3 � 2�0� � 3 �?

4�0�3 � 2�0� � 11

x � 0

x � 2

41 � 25

40 � 4 � 3 �?

32 � 4 � 11

5 � 8 � 4 � 3 �?

4�8� � 4 � 11

5�2�3 � 2�2� � 3 �?

4�2�3 � 2�2� � 11

x � 2

5x3 � 2x � 3 � 4x3 � 2x � 11

(b)

is a solution.

(d)

is not a solution.x � 10

5017 � 4009

5000 � 20 � 3 �?

4000 � 20 � 11

5�1000� � 20 � 3 �?

4�1000� � 20 � 11

5�10�3 � 2�10� � 3 �?

4�10�3 � 2�10� � 11

x � 10

x � �2

�47 � �47

�40 � 4 � 3 �?

�32 � 4 � 11

5��8� � 4 � 3 �?

4��8� � 4 � 11

5��2�3 � 2��2� � 3 �?

4��2�3 � 2��2� � 11

x � �2

Page 12: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

78 Chapter 1 Equations, Inequalities, and Mathematical Modeling

5.

(a)

is a solution.x � �12

3 � 3

�5 � 8 �?

3

52��1�2� �

4��1�2� �

?3

52x

�4x

� 3

(c) is undefined.

is not a solution.x � 0

52�0� �

40

(b)

is not a solution.

(d)

is not a solution.x �14

�6 � 3

10 � 16 �?

3

52�1�4� �

41�4

�?

3

x � 4

�38

� 3

58

� 1 �?

3

52�4� �

44

�?

3

7.

(a)

is not a solution.x � 3

�7 � 4

�3�3� � 2 �?

4

�3x � 2 � 4

(b)

is not a solution.x � 2

�4 � 4

�3�2� � 2 �?

4 (c)

is not a solution.x � 9

�25 � 4

�3�9� � 2 �?

4 (d)

is not a solution.x � �6

��20 � 4

�3��6� � 2 �?

4

6.

(a)

is a solution.x � �1

4 � 4

3 � 1 �?

4

3 �1

1�?

4

3 �1

�1 � 2�?

4

x � �1

3 �1

x � 2� 4

(b)

Division by zero is undefined.

is not a solution.x � �2

3 �1

0�?

4

3 �1

�2 � 2�?

4

x � �2 (c)

is not a solution.x � 0

312 � 4

3 �1

0 � 2�?

4

x � 0 (d)

is not a solution.x � 5

317 � 4

3 �1

5 � 2�?

4

x � 5

8.

(a)

is not a solution.x � 2

3��6 � 3

3�2 � 8 �?

3

x � 2

3�x � 8 � 3

(b)

is not a solution.x � �5

3��13 � 3

3��5 � 8 �?

3

x � �5 (c)

is a solution.x � 35

3 � 3

3�27 �?

3

3�35 � 8 �?

3

x � 35 (d)

is not a solution.x � 8

0 � 3

3�8 � 8 �?

3

x � 8

9.

(a)

is a solution.

(c)

is a solution.x �72

0 � 0

1472 �

772 �

702 �

?0

6�72�2

� 11�72� � 35 �

?0

x � �53

0 � 0

503 �

553 �

1053 �

?0

6��53�2

� 11��53� � 35 �

?0

6x2 � 11x � 35 � 0

(b)

is not a solution.

(d)

is not a solution.x �53

�1103 � 0

503 �

553 �

1053 �

?0

6�53�2

� 11�53� � 35 �

?0

x � �27

�153749 � 0

2449 �

15449 �

171549 �

?0

6��27�2

� 11��27� � 35 �

?0

Page 13: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.2 Linear Equations in One Variable 79

10.

(a)

is a solution.

(c)

is not a solution.x � �13

�1439 � 0

109 �

639 �

909 �

?0

10�19� � 7 � 10 �

?0

10��13 �2

� 21��13 � � 10 �

?0

x � �13

x �25

0 � 0

85 �425 �

505 �

?0

10� 425� �

425 � 10 �

?0

10�25 �2

� 21�25 � � 10 �

?0

x �25

10x2 � 21x � 10 � 0

(b)

is a solution.

(d)

is not a solution.x � �2

�12 � 0

40 � 42 � 10 �?

0

10�4� � 42 � 10 �?

0

10��2�2 � 21��2� � 10 �?

0

x � �2

x � �52

0 � 0

1252 �

1052 �

202 �

?0

10�254 � �

1052 � 10 �

?0

10��52 �2

� 21��52 � � 10 �

?0

x � �52

11. is an identity by the DistributiveProperty. It is true for all real values of x.2�x � 1� � 2x � 2 12. is conditional. There are real

values of x for which the equation is not true (for example, x � 0).

3�x � 2� � 5x � 4

13. is conditional. There arereal values of x for which the equation is not true.�6�x � 3� � 5 � �2x � 10 14. is an identity by simplification.

It is true for all real values of x.

3�x � 2� � 5 � 3x � 6 � 5 � 3x � 1

3�x � 2� � 5 � 3x � 1

15.

This is an identity by simplification. It is true for all realvalues of x.

4�x � 1� � 2x � 4x � 4 � 2x � 2x � 4 � 2�x � 2� 16. is an identity by simplification. It is true for all real values of x.

� 21 � 3x � 3�7 � x�

�7�x � 3� � 4x � �7x � 21 � 4x

�7�x � 3� � 4x � 3�7 � x�

17.

Thus, is an identity bysimplification. It is true for all real values of x.

x2 � 8x � 5 � �x � 4�2 � 11

�x � 4�2 � 11 � x2 � 8x � 16 � 11 � x2 � 8x � 5 18. is an identity by simplification. It is true for all real values of x.x2 � 2�3x � 2� � x2 � 6x � 4

19. is conditional. There are real values

of x for which the equation is not true.

3 �1

x � 1�

4xx � 1

20. is conditional. There are real values of x for

which the equation is not true (for example, x � 1).

5

x�

3

x� 24

21. Original equation

Subtract 32 from both sides.

Simplify.

Divide both sides by 4.

Simplify. x �514

4x4

�514

4x � 51

4x � 32 � 32 � 83 � 32

4x � 32 � 83 22. Original equation

Remove parentheses.

Simplify.

Add 2 to both sides.

Simplify.

Divide both sides by 3.

Simplify. x � 3

3x

3�

9

3

3x � 9

3x � 2 � 2 � 7 � 2

3x � 2 � 7

3x � 12 � 10 � 7

3�x � 4� � 10 � 7

Page 14: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

80 Chapter 1 Equations, Inequalities, and Mathematical Modeling

23.

x � 4

x � 11 � 11 � 15 � 11

x � 11 � 15 24.

�12 � x

7 � 19 � 19 � x � 19

7 � 19 � x

7 � x � x � 19 � x

7 � x � 19 25.

x � �9

�2x�2

�18�2

�2x � 18

7 � 7 � 2x � 25 � 7

7 � 2x � 25

26.

x � 3

7x7

�217

7x � 21

7x � 2 � 2 � 23 � 2

7x � 2 � 23 27.

x � 5

5x5

�255

5x � 25

5x � 5 � 5 � 20 � 5

5x � 5 � 20

8x � 3x � 5 � 3x � 3x � 20

8x � 5 � 3x � 20 28.

x � �5

4x � �20

7x � 3 � 3 � 3x � 3x � 17 � 3 � 3x

7x � 3 � 3x � 17

29.

x � 9

�x � �9

�x � 3 � 3 � �6 � 3

�x � 3 � �6

2x � 3x � 3 � 3x � 3x � 6

2x � 3 � 3x � 6

2x � 10 � 7 � 3x � 6

2�x � 5� � 7 � 3�x � 2� 30.

x � �58

8x � �5

3x � 9 � 5x � 9 � 4 � 5x � 5x � 9

3x � 9 � 4 � 5x

3x � 9 � 5 � 5x � 1

3�x � 3� � 5�1 � x� � 1 31.

No solution

�9 � 8

�5x � 5x � 9 � 8 � 5x � 5x

�5x � 9 � 8 � 5x

x � 6x � 9 � 8 � 5x

x � 3�2x � 3� � 8 � 5x

32.

The solution is the set of all realnumbers.

9x � 10 � 9x � 10

9x � 10 � 5x � 4x � 10

9x � 10 � 5x � 2�2x � 5� 33.

x � �4

5x � 2 � 4x � 2

4�5x4 � � 4�1

2� � 4�x� � 4�12�

4�5x4

�12� � 4�x �

12�

5x4

�12

� x �12

34.

x � �5

�6x � 30

2x � 5x � 30 � 3x

10�x5

�x2� � 10�3 �

3x10�

x5

�x2

� 3 �3x10

35.

z � �65

5z � �6

6z � 30 � z � 24 � 0

6�z � 5� � �z � 24� � 0

4�32��z � 5� � 4�1

4��z � 24� � 4�0�

4�32

�z � 5� �14

�z � 24� � 4�0�

32

�z � 5� �14

�z � 24� � 0 36.

x � 6

7x � 42

7x � 2 � 40

6x � �x � 2� � 40

4�3x

2 � � 4�1

4��x � 2� � 4�10�

4�3x

2�

1

4�x � 2� � 4�10�

3x

2�

1

4�x � 2� � 10

Page 15: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.2 Linear Equations in One Variable 81

37.

x � 9

�0.50x � �4.5

�0.50x � 7.5 � 3

0.25x � 7.5 � 0.75x � 3

0.25x � 0.75�10 � x� � 3 38.

x � 50

0.20x � 10

0.60x � 40 � 0.40x � 50

0.60x � 0.40�100 � x� � 50

39. or

The second way is easier since you are not working withfractions until the end of the solution.

x �73 x �

73

3x � 7 x �43 � 1

3x � 3 � 4 x � 1 �43

3�x � 1� � 4 3�x � 1� � 4 40. or

The second way is easier because you are not working with fractions until the end of the solution.

x �34 x �

34

4x � 3 x �154 � 3

4x � 12 � 15 x � 3 �154

4�x � 3� � 15 4�x � 3� � 15

41. or

The first way is easier. The fraction is eliminated in thefirst step.

x � 13

x � 3�133 �

13x �133 x � 13

13x � 5 �23 x � 2 � 15

13x �23 � 53�1

3��x � 2� � 3�5�

13�x � 2� � 5 13�x � 2� � 5 42. or

The first way is easier because the fraction is eliminated in the first step.

z � 12

43 � 34z �

43 � 9 z � 12

34z � 9 z � 4 � 8

34z � 3 � 6 �43�3

4�z � 4� � �43�6

34�z � 4� � 6 34�z � 4� � 6

43.

Contradiction; no solution

8 � �4

x � 8 � x � 4

x � 8 � 2x � 4 � x

x � 8 � 2�x � 2� � x 44.

Contradiction; no solution

13 � 10

2x � 13 � 2x � 10

8x � 16 � 6x � 3 � 2x � 10

8�x � 2� � 3�2x � 1� � 2�x � 5�

45.

x � 10

�31x � �310

400 � 16x � 15x � 18 � 72

4�100 � 4x� � 3�5x � 6� � 72

12�100 � 4x3 � � 12�5x � 6

4 � � 12�6�

100 � 4x

3�

5x � 64

� 6 46.

1

2� y

49 � 98y

49 � 2y � 100y

17 � y � 32 � y � 100y

�y�17 � y

y� �y�

32 � y

y� 100�y�

17 � y

y�

32 � y

y� 100

Page 16: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

82 Chapter 1 Equations, Inequalities, and Mathematical Modeling

47.

x � 4

5x � 20

15x � 12 � 10x � 8

3�5x � 4� � 2�5x � 4�

5x � 45x � 4

�23

48.

x � 0

15x � 0

20x � 6 � 5x � 6

2�10x � 3� � 1�5x � 6�

10x � 3

5x � 6�

1

2

49.

x � 3

6x � 18

10x � 13 � 4x � 5

10x � 13

x�

4x � 5x

10 �13x

� 4 �5x

50.

9

7� x

9 � 7x

9

x� 7

15

x�

6

x� 7

15

x� 4 �

6

x� 3 51.

z � 0

3z � 6 � 2z � 4 � 2

3�z � 2� � �2 �2

z � 2��z � 2�

3 � 2 �2

z � 2

52. Multiply both sides by

x �53

3x � 5

3x � 5 � 0

1�x � 5� � 2x � 0

x�x � 5�. 1x

�2

x � 5� 0 53.

Contradiction; no solution

3 � 0

1 � 2 � 0

x � 4x � 4

� 2 � 0

x

x � 4�

4x � 4

� 2 � 0

54. Multiply both sides by

x �116

6x � 11

14x � 7 � 16x2 � 8x � �16x2 � 4

7�2x � 1� � 8x�2x � 1� � �4�2x � 1��2x � 1�

�2x � 1��2x � 1�. 7

2x � 1�

8x2x � 1

� �4

55. Multiply both sides by

A check reveals that is an extraneous solution—it makes the denominator zero. There is no real solution.x � 4

4 � x

12 � 3x

2 � 3x � 10

2 � x � 2 � 2x � 8

2 � 1�x � 2� � 2�x � 4�

�x � 4��x � 2�. 2

�x � 4��x � 2� �1

x � 4�

2x � 2

Page 17: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.2 Linear Equations in One Variable 83

56. Multiply both sides by

x � �133

3x � �13

18x � 2 � 15x � 15

12x � 4 � 6x � 6 � 15x � 15

4�3x � 1� � 6�x � 1� � 15�x � 1�

�x � 1��3x � 1� 4x � 1

� �x � 1��3x � 1� 63x � 1

� �x � 1��3x � 1� 153x � 1

�x � 1��3x � 1�. 4

x � 1�

63x � 1

�15

3x � 1

57.

Multiply both sides by

x � 5

2x � 10

1�x � 3� � 1�x � 3� � 10

�x � 3��x � 3�. 1

x � 3�

1x � 3

�10

�x � 3��x � 3�

1

x � 3�

1x � 3

�10

x2 � 9

58.

Multiply both sides by

x �7

4

4x � 7

4x � 3 � 4

x � 3 � 3x � 6 � 4

�x � 3� � 3�x � 2� � 4

�x � 3��x � 2�.1

x � 2�

3

x � 3�

4

�x � 3��x � 2�

1

x � 2�

3

x � 3�

4

x2 � x � 6

59.

Multiply both sides by

A check reveals that is an extraneous solution since it makes the denominator zero, so there is no solution.x � 3

x � 3

3x � 9

3 � 4x � 12 � x

3 � 4�x � 3� � x

x�x � 3�.3x�x � 3� �

4x

�1

x � 3

3

x2 � 3x�

4x

�1

x � 3

60. Multiply both sides by

x � �3

4x � 18 � 3x � 15

6x � 18 � 2x � 3x � 15

6�x � 3� � 2x � 3�x � 5�

x�x � 3�. 6

x�

2

x � 3�

3�x � 5�x�x � 3�

Division by zero is undefined. Thus, is not asolution, and the original equation has no solution.

x � �3

�2 �2

0�

�6

�3�0�

Check: 6

�3�

2

�3 � 3�

3��3 � 5��3��3 � 3�

Page 18: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

84 Chapter 1 Equations, Inequalities, and Mathematical Modeling

61.

x � 0

�2x � 0

4x � 9 � 6x � 9

x2 � 4x � 4 � 5 � x2 � 6x � 9

�x � 2�2 � 5 � �x � 3�2 62.

x �15

5x � 1

x2 � 2x � 1 � 2x � 4 � x2 � x � 2

�x � 1�2 � 2�x � 2� � �x � 1��x � 2�

63.

The equation is an identity; every real number is a solution.

4 � 4

x2 � 4x � 4 � x2 � 4x � 4

�x � 2�2 � x2 � 4�x � 1� 64.

This is a contradiction. Thus, the equation has no solution.

1 � 4

4x2 � 4x � 1 � 4x2 � 4x � 4

�2x � 1�2 � 4�x2 � x � 1�

65.

The x-intercept is at 3. The solution to and the x-intercept of are the same.They are both The x-intercept is �3, 0�.x � 3.

y � 2�x � 1� � 40 � 2�x � 1� � 4

x � 3

3 � x

6 � 2x

0 � 2x � 6

0 � 2x � 2 � 4

−6 12

−8

4

0 � 2�x � 1� � 4y � 2�x � 1� � 4 66.

Intercept:

The solution to is the same as the x-intercept of They are both x � �

32.y �

43x � 2.

0 �43x � 2

��32, 0�

x � � 32

�� 34��� 4

3 x� � �� 34��2�

�43x � 2

0 �43x � 2

−5 4

−3

3

y �43x � 2

67.

The x-intercept is at 10. The solution toand the x-intercept of are the same. They are both The x-intercept is �10, 0�.x � 10.

y � 20 � �3x � 10�0 � 20 � �3x � 10�

x � 10

3x � 30

0 � 30 � 3x

0 � 20 � 3x � 10

−20 40

35

−5

0 � 20 � �3x � 10�y � 20 � �3x � 10� 68.

Intercept:

The solution to is the same as the x-intercept of They are both x � �3.y � 10 � 2�x � 2�.

0 � 10 � 2�x � 2�

��3, 0�

x � �3

�2x � 6

0 � 6 � 2x

0 � 10 � 2x � 4

0 � 10 � 2�x � 2�

−12 6

−2

10

y � 10 � 2�x � 2�

69.

The x-intercept is at The solution to and the x-intercept of are the same.They are both The x-intercept is �7

5, 0�.x �75.

y � �38 � 5�9 � x�0 � �38 � 5�9 � x�7

5.

x �75

5x � 7

0 � 7 � 5x

0 � �38 � 45 � 5x

−6 10

−2

10

0 � �38 � 5�9 � x�y � �38 � 5�9 � x� 70.

There is no x-intercept.

0 � �9611

0 � 6x �9611 � 6x

0 � 6x � 6�1611 � x�

−8 10

−10

2

y � 6x � 6�1611 � x�

Page 19: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.2 Linear Equations in One Variable 85

72.

The x-intercept is the y-intercept is �0, 16�.�163 , 0�;

163 � x

y � 16 �16 � �3x

y � 16 � 3�0� 0 � 16 � 3x

y � 16 � 3x y � 16 � 3x

73.

x-intercept:

y-intercept:

The x-intercept is and the y-intercept is �0, �3�.��12, 0�

y � �3�2�0� � 1� ⇒ y � �3

0 � �3�2x � 1� ⇒ 0 � 2x � 1 ⇒ x � �12

y � �3�2x � 1� 74.

The x-intercept is the y-intercept is �0, �1�.�1, 0�;

1 � x

y � �1 0 � �1 � x

y � 5 � �6 � 0� 0 � 5 � �6 � x�

y � 5 � �6 � x� y � 5 � �6 � x�

75.

x-intercept:

y-intercept:

The x-intercept is and the y-intercept is �0, 103 �.�5, 0�

2�0� � 3y � 10 ⇒ 3y � 10 ⇒ y �103

2x � 3�0� � 10 ⇒ 2x � 10 ⇒ x � 5

2x � 3y � 10 76.

The x-intercept is the y-intercept is �0, �125 �.�3, 0�;

y � �125 x � 3

�5y � 12 4x � 12

4�0� � 5y � 12 4x � 5�0� � 12

4x � 5y � 12 4x � 5y � 12

77. Multiply both sides by 5.

x-intercept:

y-intercept:

The x-intercept is and the y-intercept is �0, 83�.��20, 0�

2�0� � 40 � 15y � 0 ⇒ 40 � 15y � 0 ⇒ y �4015

�83

2x � 40 � 15�0� � 0 ⇒ 2x � 40 � 0 ⇒ x � �20

2x5

� 8 � 3y � 0 ⇒ 2x � 40 � 15y � 0

78.

The x-intercept is the y-intercept is �0, 52�.��158 , 0�;

x � �158

38

�8x3

�38

� ��5�

8x3

� �5

8x3

� 5 � 2�0� � 0

8x3

� 5 � 2y � 0

y �52

�2y � �5

8�0�

3� 5 � 2y � 0

8x3

� 5 � 2y � 0

79.

x-intercept:

y-intercept:

The x-intercept is and the y-intercept is �0, �0.3�.�1.6, 0�

4y � 0.75�0� � 1.2 � 0 ⇒ 4y � 1.2 � 0 ⇒ y ��1.2

4� �0.3

4�0� � 0.75x � 1.2 � 0 ⇒ � 0.75x � 1.2 � 0 ⇒ x �1.2

0.75� 1.6

4y � 0.75x � 1.2 � 0

71.

x-intercept:

y-intercept:

The x-intercept is and the y-intercept is �0, 12�.�125 , 0�

y � 12 � 5�0� ⇒ y � 12

0 � 12 � 5x ⇒ 5x � 12 ⇒ x �125

y � 12 � 5x

Page 20: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

86 Chapter 1 Equations, Inequalities, and Mathematical Modeling

80.

The x-intercept is the y-intercept is �0, 1.13�.�1.36, 0�;

x � 1.36

x �3.42.5

2.5x � 3.4

3�0� � 2.5x � 3.4 � 0

3y � 2.5x � 3.4 � 0

y � 1.13

y �3.43

3y � 3.4

3y � 2.5�0� � 3.4 � 0

3y � 2.5x � 3.4 � 0

81.

x �1

3 � a, a � 3

x�3 � a� � 1

3x � ax � 1

4x � 4 � ax � x � 5

4�x � 1� � ax � x � 5 82.

1 � 4b

2 � a� x, a � �2

1 � 4b � x�2 � a�

1 � 4b � 2x � ax

4 � 2x � 4b � ax � 3

4 � 2�x � 2b� � ax � 3

83.

x �5

4 � a, a � �4

x�4 � a� � 5

4x � ax � 5

6x � ax � 2x � 5 84.

x �7

a � b, a � �b

x�a � b� � 7

ax � bx � 7

5 � ax � 12 � bx

85.

Multiply both sides by 2.

x �18

36 � a, a � �36

x�36 � a� � 18

36x � ax � 18

18x �12

ax � 9

19x �12

ax � x � 9 86.

x �30b � 43a � 15

, a � �5

x�15 � 3a� � 30b � 4

�15x � 30b � 12 � 8 � 3ax

�5�3x � 6b� � 12 � 8 � 3ax

87.

x ��17

�2a � 10�

�1710 � 2a

, a � 5

x��2a � 10� � �17

�2ax � 10x � �17

�2ax � 10x � 18 � 1

�2ax � 6x � 18 � �4x � 1

�2ax � 6�x � 3� � �4x � 1 88.

x � �8a

, a � 0

ax � �8

45

x � ax �45

x � 2 � 10

45

x � ax � 2�25

x � 1� � 10

89.

138.889

x �62.50.45

�0.45x � �62.5

0.275x � 362.5 � 0.725x � 300

0.275x � 0.725�500 � x� � 300 90.

1.326 x

20.9074 � 15.77x

2.763 � 9.45x � 23.1444 � 6.32x � 5

2.763 � 4.5�2.1x � 5.1432� � 6.32x � 5

Page 21: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.2 Linear Equations in One Variable 87

91. Multiply both sides by

x ��5.405��7.398�

2 19.993

2x � �5.405��7.398�

2x � �4.405��7.398� � 7.398

2x � �4.405��7.398� � 7.398

7.398x. 2

7.398�

4.405x

�1x

92. Multiply both sides by

�0.342 x

�38.1 � 111.3x

3x � 38.1 � 114.3x

3x � 6�6.350� � 18�6.350�x

6.350x. 3

6.350�

6

x� 18

93.

h � 10 feet

h �471 � 50�

10��

471 � 50(3.14)10(3.14)

� 10

471 � 50� � 10�h

471 � 50� � 10�h

471 � 2��25� � 2��5h� 94.

x � 10 centimeters

200 � 20x

248 � 48 � 8x � 12x

248 � 2�24� � 2�4x� � 2�6x�

95. (a) Female:

inches

(c)

The lengths of the male and female femurs areapproximately equal when the lengths are 100 inches.

x 61.2

26.440.432

� x

26.44 � 0.432x

For y � 16: 16 � 0.432x � 10.44

y � 0.432x � 10.44 (b) Male:

Yes, it is likely that both bones came from the sameperson because the estimated height of a male with a 19-inch thigh bone is 69.4 inches.

(d)

inches

It is unlikely that a female would be over 8 feet tall,so if a femur of this length was found, it most likelybelonged to a very tall male.

x 100.59

1.71 � 0.017x

0.432x � 10.44 � 0.449x � 12.15

69.4 x

31.15 � 0.449x

For y � 19: 19 � 0.449x � 12.15

y � 0.449x � 12.15

Height x Female Femur Male FemurLength Length

60 15.48 14.79

70 19.80 19.28

80 24.12 23.77

90 28.44 28.26

100 32.76 32.75

110 37.08 37.24

96. (a)

(b)

The earned income E is $6800.

E � 6800

12E � 3400

For S � 6600: 6600 � 10,000 �12E

S � 10,000 �12E

T � 10,000 �12E, 0 ≤ E ≤ 20,000

T � E � �10,000 �12E�

T � E � S (c)

Earned income: $7600

(d)

For

The subsidy is $7500.

� 10,000 � 2500 � 7500

S � 10,000 �12E � 10,000 �

12�5000�

5000 � E

2500 �12E

12,500 � 10,000 �12ET � 12,500:

T � 10,000 �12E

7600 � E

3800 �12E

13,800 � 10,000 �12E

Page 22: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

88 Chapter 1 Equations, Inequalities, and Mathematical Modeling

97.

(a)

The intercept is �0, 36.8�.y-

y

t20 4 6 8 10 12

32

40

44

48

52

56

60

Year (0 ↔ 1990)

Con

sum

ptio

n of

gas

(in

billi

ons

of g

allo

ns)

y � 1.64t � 36.8, �1 ≤ t ≤ 12 (b) Let

intercept:

(c)

This corresponds with the year 2017. Explanations will vary.

t �28.21.64

17.2

28.2 � 1.64t

65 � 1.64t � 36.8

�0, 36.8�y-

y � 1.64�0� � 36.8 � 36.8t � 0:

98. (a) The number reached 33 million in 1995.

(b) Part (a) can be solved graphically by graphing and on the same axes and finding the intersection, Part (a) can be solved algebraically by solving for t.0.39t � 31.0 � 33t � 5 ⇒ 1995.

y � 33y � 0.39t � 31.0

99.

m � 23,437.5 miles

75000.32

� m

7500 � 0.32m

10,000 � 0.32m � 2500 100.

t � 28 hours

0.25t � 7

1 � �0.25t � 8

y � �0.25t � 8

101. False.

This is a quadratic equation. The equation cannot be written in the form ax � b � 0.

x�3 � x� � 10 ⇒ 3x � x2 � 10 102. False. If both sides of the equation are graphed, you cansee that they intersect, which means the equation has areal solution.

103. Equivalent equations are derived from the substitution principle and simplification techniques. They have the same solution(s).

and are equivalent equations.2x � 52x � 3 � 8

104. To transform an equation into an equivalent equation,you should first remove symbols of grouping, combinelike terms, and reduce fractions. Then, as needed, youmay add (or subtract) the same quantity to (from) bothsides of the equation, multiply (divide) both sides ofthe equation by the same nonzero quantity, or inter-change the two sides of the equation.

105. (a)

(b) Since the sign changes from negative at 1 to positive at 2, the root is somewhere between 1 and 2.

(c)

(d) Since the sign changes from negative at to positive at the root is somewhere between and

To improve accuracy, evaluate the expression at subintervals within this interval and determine where the sign changes.

1.8 < x < 1.9

1.9.1.81.9,1.8

1 < x < 2

x 0 1 2 3 4

73.80.6�2.6�5.8�93.2x � 5.8

�1

x 2

0.60.28�0.04�0.36�0.68�13.2x � 5.8

1.91.81.71.61.5

Page 23: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.2 Linear Equations in One Variable 89

106.

The solution to is in the interval

The solution is in the interval

The solution is in the interval 8.16 < x < 8.17.

8.1 < x < 8.2.

8 < x < 9.0.3�x � 1.5� � 2 � 0

0.3�x � 1.5� � 2 � 0

x 6 7 8 9 10

�0.65 �0.35 �0.05 0.25 0.550.3�x � 1.5� � 2

x 8.1 8.2 8.3 8.4

�0.02 0.01 0.04 0.070.3�x � 1.5� � 2

x 8.14 8.15 8.16 8.17 8.18

�0.008 �0.005 �0.002 0.001 0.0040.3�x � 1.5� � 2

107.x2 � 5x � 36

2x2 � 17x � 9�

�x � 9��x � 4��2x � 1��x � 9� �

x � 42x � 1

, x � �9 108.

cannot be simplified.

x2 � 49x3 � x2 � 3x � 21

109.

Intercepts: �0, �5�, �53, 0�

x−3 −2 −1 1 2 3 4 5

−1

1

2

−2

−3

−4

−5

y

y � 3x � 5 110.

Intercepts: �0, �92�, ��9, 0�

x2

−6

4

4

−2

−8

2

−10

−2−4

y

y � �12x �

92

111.

Intercepts: �0, 0�, ��5, 0�

x1

1

4

5

6

7

2−7 −6 −4 −3 −2 −1−1

−2

y

y � �x2 � 5x � �x�x � 5� 112.

Intercepts: �5, 0�, �0, �5�

x

6

−1

1

−2

−1−2 1 2 3 4 5 6

2

3

4

5

y

y � �5 � x

Page 24: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.3 Modeling with Linear Equations

90 Chapter 1 Equations, Inequalities, and Mathematical Modeling

■ You should be able to set up mathematical models to solve problems.

■ You should be able to translate key words and phrases.

(a) Equality: (b) Addition:

Equals, equal to, is, are, was, will be, represents Sum, plus, greater, increased by, more than,exceeds, total of

(c) Subtraction: (d) Multiplication:

Difference, minus, less than, decreased by, Product, multiplied by, twice, times, percent ofsubtracted from, reduced by, the remainder

(e) Division: (f) Consecutive:

Quotient, divided by, ratio, per Next, subsequent

■ You should know the following formulas:

(a) Perimeter: (b) Area:

1. Square: 1. Square:

2. Rectangle: 2. Rectangle:

3. Circle: 3. Circle:

4. Triangle: 4. Triangle:

(c) Volume

1. Cube:

2. Rectangular solid:

3. Cylinder:

4. Sphere:

(d) Simple Interest: (e) Compound Interest:

(f) Distance: (g) Temperature:

■ You should be able to solve word problems. Study the examples in the text carefully.

F �9

5 C � 32d � rt

A � P�1 �r

n�nt

I � Prt

V � �4

3��r 3

V � �r 2 h

V � lwh

V � s3

A � �1

2�bhP � a � b � c

A � �r2C � 2�r

A � lwP � 2l � 2w

A � s2P � 4s

1.

The sum of a number and 4 A number increased by 4

x � 4 2.

A number decreased by 10The difference of a number and 10

t � 10 3.

The ratio of a number and 5The quotient of a number and 5A number divided by 5

u

5

Vocabulary Check

1. mathematical modeling 2. verbal model; 3. 4.algebraic equation

5. 6. 7. 8. I � PrtA � P�1 �r

12�12t

V � �r2hV � s3

P � 2l � 2wA � �r2

Page 25: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.3 Modeling with Linear Equations 91

4.

The product of and a number Two-thirds of a number

23

23

x 5.

The difference of a number and 4 is divided by 5. A number decreased by 4 is divided by 5.

y � 4

56.

The sum of a number and 10 isdivided by 7. A number increased by 10 is divid-ed by 7.

z � 107

7.

The product of and the sum of a number and 2Negative 3 is multiplied by a number increased by 2.

�3

�3�b � 2� 8.

The product of and thedifference of a number and 1 is divided by 8. Negative 5 is multiplied by anumber decreased by 1 and isdivided by 8.

�5

�5�x � 1�8

9.

The difference of a number and 5 is multiplied by 12 times thenumber.12 is multiplied by a number andthat product is multiplied by thenumber decreased by 5.

12x�x � 5�

10.

The product of the sum of a number and 4 and the difference of 3 and a number is divided by the product of 2 and a number.A number increased by 4 is multiplied by 3 decreased by a number and the product is divided by 2 multiplied by a number.

�q � 4��3 � q�2q

11. Verbal Model: (Sum) (first number) (second number)

Labels: Sum first number second number

Expression: S � n � �n � 1� � 2n � 1

� n � 1� n,� S,

��

12. Verbal Model: Product (first number) (second number)

Labels: Product first number second number

Expression: P � n�n � 1� � n2 � n

� n � 1� n,� P,

��

13. Verbal Model: Product (first odd integer)(second odd integer)

Labels: Product first odd integer second odd integer

Expression: P � �2n � 1��2n � 1� � 4n2 � 1

� 2n � 1 � 2 � 2n � 1� 2n � 1,� P,

14. Verbal Model: (Sum) (first even number) (second even number)

Labels: Sum first even number second even number

Expression: S � �2n�2 � �2n � 2�2 � 4n2 � 4n2 � 8n � 4 � 8n2 � 8n � 4

� 2n � 2� 2n,� S,

22 ��

15. Verbal Model: (Distance) (rate) (time)

Labels: Distance rate time

Expression: d � 50t

� t� 50 mph,� d,

�� 16. Verbal Model: (Distance) (rate) (time)

Labels: Distance km, rate time

Expression: t � 200�r200 � rt,

� t� r,� 200

��

17. Verbal Model: (Amount of acid) (amount of solution)

Labels: Amount of acid (in gallons) amount of solution (in gallons)

Expression: A � 0.20x

� x� A,

� 20% �

Page 26: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

92 Chapter 1 Equations, Inequalities, and Mathematical Modeling

18. Verbal Model: (Sale price) (list price) (discount)

Labels: Sale price list pricediscount

Expression: S � L � 0.2L � 0.8L

� 0.2L� L,� S,

�� 19. Verbal Model:

Labels: Perimeter widthlength (width)

Expression: P � 2x � 2�2x� � 6x

� 2x� 2� x,� P,

Perimeter � 2�width� � 2�length�

20. Verbal Model: (Area) (height)

Labels: Area base 20 inches, height inches

Expression: A �12�20�h � 10h

� h�� A,

�12(base)

21. Verbal Model: (Total cost) (unit cost)(number of units) (fixed cost)

Labels: Total cost fixed cost unit cost number of units

Expression: C � 25x � 1200

� x� $25,� $1200,� C,

��

22. Verbal Model: (Revenue) (price)(number of units)

Labels: Revenue price 3.59, number of units

Expression: R � 3.59x

� x�� R,

23. Verbal Model: Thirty percent of the list price L

Expression: 0.30L

24. Verbal Model: 35% of q

Expression: 0.35q

25. Verbal Model: percent of 500 that is represented by thenumber N

Equation: , p is in decimal formN � p�500�

26. as a percent of

�S2 � S1�S1

�100�

S1�S2 � S1�

27.

A � 4x � 8x � 12x

Area � Area of top rectangle � Area of bottom rectangle

4

42x

x

x

8

28.

Area �12b�2

3b � 1� �13b2 �

12b

Area �12�base��height�

29. Verbal Model: Sum (first number) (second number)

Labels: Sum first number second number

Equations:

Answer: First number second number � n � 1 � 263� n � 262,

n � 262

524 � 2n

525 � 2n � 1

525 � n � �n � 1�

� n � 1� n,� 525,

��

Page 27: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.3 Modeling with Linear Equations 93

30. Verbal Model: (Sum) (first number) (second number) (third number)

Labels: Sum first number second number third number

Equation:

Answer: (second number), and (third number)n � 2 � 269n � 267, n � 1 � 268

267 � n

801 � 3n

804 � 3n � 3

804 � n � n � 1 � n � 2

� n � 2� n � 1,� n,� 804,

���

31. Verbal Model: Difference (one number) (another number)

Labels: Difference one number another number

Equation:

Answer: The two numbers are 37 and 185.

5x � 185

x � 37

148 � 4x

148 � 5x � x

� x� 5x,� 148,

��

32. Verbal Model: (Difference) (number) (one-fifth of number)

Labels: Difference number one-fifth of number

Equation:

Answer: The numbers are 95 and 15 � 95 � 19.

95 � n

76 �45n

76 � n �15n

�15n� n,� 76,

��

33. Verbal Model: Product (smaller number) (larger number)

Labels: Smaller number larger number

Equation:

Answer: Smaller number larger number � n � 1 � �4� n � �5,

n � �5

n2 � n � n2 � 5

n�n � 1� � n2 � 5

� n � 1� n,

� �smaller number)2 � 5��

34. Verbal Model: Difference (reciprocal of smaller number) (reciprocal of larger number) (reciprocal of smaller number)

Labels: Smaller number larger number difference

Equation: Multiply both sides by

Answer: The numbers are 3 and n � 1 � 4.

n � 3

n � 1 � 4n � 4 � 4n

n � 1 � 4�n � 1� � 4n

4n�n � 1�1

4n� 4n�n � 1�

1

n� 4n�n � 1�

1

n � 1

4n�n � 1�. 1

4n�

1

n�

1

n � 1

�1

4n� n � 1,� n,

�14

��

Page 28: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

94 Chapter 1 Equations, Inequalities, and Mathematical Modeling

35.

� 13.5

� 0.30�45�

� �30%��45�

x � percent � number 36.

� 630

� �1.75��360�

� �175%��360�

x � percent � number 37.

p � 0.27 � 27%

4321600 � p

432 � p�1600�

432 � percent � 1600

38.

p � 1.35 � 135%

459

340� p

459 � p�340�

459 � percent � 340 39.

x � 2400

12

0.005� x

12 � 0.005x

12 �12

% � number 40.

number �70

40%�

70

0.40� 175

70 � 40% � number

41. Verbal Model: Loan payments

Labels: Loan payments

Annual income

Equation:

The family’s annual income is $22,316.98.

I � 22,316.98

13,077.75 � 0.586I

� I

� 13,077.75

� 58.6% � Annual income

42. (a) Total income billion

Corporate taxes:

Income tax:

Social Security:

Other:

(b)

Interest on debt:

Health and human services:

Defense department:

Other:

(c) There is a deficit of billion.1853.2 � 2011.0 � �$157.8

173.52011

� 0.0863 � 8.6%

348.62011

� 0.1734 � 17.3%

1317.92011

� 0.65535 � 65.5%

1712011

� 0.085 � 8.5%

Defense Department17.3%

Interest on debt8.5%

Other 8.6%

Health and HumanServices 65.5%

Total expenses � 171.0 � 1317.9 � 348.6 � 173.5 � $2011 billion

146.11853.2

� 0.079 or 7.9%

700.81853.2

� 0.378 or 37.8%

858.31853.2

� 0.463 or 46.3%

Other7.9%

Corporation taxes8.0%

Social Security37.8%

Income tax46.3%

148.01853.2

� 0.080 or 8%

� 148.0 � 858.3 � 700.8 � 146.1 � $1853.2

Page 29: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.3 Modeling with Linear Equations 95

44. Verbal Model:

Labels:

Equation:

The percent increase in the price of a half-gallon of ice cream from 1990 to 2002 is 48%.

p �1.222.54

� 0.48

1.22 � 2.54p

3.76 � 2.54p � 2.54

1990 price � $2.54percent increase � p,2002 price � $3.76,

�2002 price� � �percent increase��1990 price� � 1990 price

45. Verbal Model:

Labels: 2002 price for a pound of tomatoes: $1.66

1990 price for a pound of tomatoes: $0.86

Percentage increase:

Equation:

Answer: Percentage increase � 93%

0.930 � p

0.80 � 0.86p

1.66 � 0.86p � 0.86

p

� �1990 price for a pound of tomatoes�

� �percentage increase��1990 price for a pound of tomatoes� �2002 price for a pound of tomatoes�

43. Verbal Model:

Labels: 2002 price for a gallon of unleaded gasoline: $1.36

1990 price for a gallon of unleaded gasoline: $1.16

Percentage increase:

Equation:

Answer: Percentage increase � 17.2%

0.172 � p

0.20 � 1.16p

1.36 � 1.16p � 1.16

p

� �1990 price for a gallon of unleaded gasoline�

� �percentage increase��1990 price for a gallon of unleaded gasoline� �2002 price for a gallon of unleaded gasoline�

46. Verbal Model:

Labels:

Equation:

decrease

The percent decrease in the price of a personal computer from 1990 to 2002 is 18.6%.

p ��1951050

� �0.1857 � 18.6%

�195 � 1050p

855 � 1050p � 1050

1990 price � $1050percent increase � p,2002 price � $855,

�2002 price� � �percent increase��1990 price� � 1990 price

47. Verbal Model: (Sale price) (list price) (discount)

Labels: Sale price list price discount

Equation:

Answer: The list price of the pool is $1450.

1450 � L

1210.75 � 0.835L

1210.75 � L � 0.165L

� 0.165L� L,� $1210.75,

��

Page 30: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

96 Chapter 1 Equations, Inequalities, and Mathematical Modeling

49. (a)

w

l

(b)

� 5w

� 2�1.5w� � 2w

P � 2l � 2w

l � 1.5w (c)

Width:

Length:

Dimensions: 7.5 meters � 5 meters

l � 1.5w � 7.5 meters

w � 5 meters

5 � w

25 � 5w

50. (a)

w

h

(b)

� 3.24w

� 2�0.62w� � 2w

P � 2h � 2w

h � 0.62w (c)

Width:

Height:

Dimensions: 0.38 m � 0.62 m

h � 0.62w � 0.38 meters

w � 0.62 meters

0.62 � w

2 � 3.24w

51. Verbal Model: Average

Labels: Average

Equation:

Answer: You must score 97 (or better) on test #4 to earn an A for the course.

x � 97

360 � 263 � x

90 �87 � 92 � 84 � x

4

test #4 � xtest #3 � 84,test #2 � 92,test #1 � 87,� 90,

��test #1� � �test #2� � �test #3� � �test #4�

4

52. Verbal Model: (Average)

Labels: Average test #1 test #2 test #3 test #4

Equation:

You must score 187 out of 200 on the last test to get an A in the course.

187 � x

450 � 263 � x

450 � 87 � 92 � 84 � x

90 �87 � 92 � 84 � x

5

� x� 84,� 92,� 87,� 90,

��test #1� � �test #2� � �test #3� � �test #4�

5

53.

Total time �total distance

rate�

300 kilometers100 kilometers�hour

� 3 hours

Rate �distance

time�

50 kilometers12 hour

� 100 kilometers�hour

48. Verbal Model: (Total) (coworker’s paycheck)

Labels: Total $645, coworker’s paycheck salesperson’s paycheck

Equation:

Coworker’s paycheck is $348.65 and salesperson’s paycheck is 0.85x � $296.35.

348.65 � x

645 � 1.85x

645 � 0.85x � x

� x � 0.15x � 0.85x� x,�

� �salesperson’s paycheck) �

Page 31: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.3 Modeling with Linear Equations 97

54.

hour (or 20 minutes) must elapse before the two cars are 5 miles apart.13

t �13 hour

5 � 55t � 40t � 15t

5 � d2 � d1

(Distance between cars) � (second distance) � (first distance)

d2 � 55 mph � t

d1 � 40 mph � t

Distance � rate � time

55.

Multiply both sides by

The salesman averaged 58 miles per hour for 3 hours and 52 miles per hour for 2 hours and 45 minutes.

t2 �317 � 174

52� 2.75 hours

t1 �17458

� 3 hours

x � 174 miles

�6x � �1044

52x � 18,386 � 58x � 17,342

52x � 58�317 � x� � 5.75�58��52�

58�52�. x

58�

317 � x52

� 5.75

d1

r1�

d2

r2� T

t1 � t2 � T

�time on first part� � �time on second part) � �Total time�

56. Verbal Model: (Distance traveled by first car) (distance traveled by second car)

Labels: Distance traveled by first car distance traveled by second car

Equation:

The second car catches up to the first car after 2.75 hours (or 2 hours and 45 minutes). The distance traveled is miles. Thus, the second car catches up before the first car arrives at the game.45�2.75� � 123.75

t � 2.75

�10t � �27.5

45t � 55t � 27.5

45t � 55�t �12�

� 55�t �12�� 45t,

57. (a) Time for the first family:

Time for the other family: hours

(b)

(c) d � r t � 42�16042

�16050 � � 25.6 miles

t �dr

�100

42 � 50�

10092

� 1.08 hours

t2 �dr2

�16050

� 3.2

t1 �dr1

�16042

� 3.81 hours

Page 32: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

98 Chapter 1 Equations, Inequalities, and Mathematical Modeling

58. Verbal Model: (Distance) (rate)(time time

Labels: Distance miles, rate

,

Equation:

The average speed for the round trip was approximately46.3 miles per hour.

46.3 � r

400 � r�1600

440�

2200

440 � �3800

440r

400 � r�200

55�

200

40 �

time2 �distance

rate2

�200

40 hours

time1 �distance

rate1

�200

55 hours

� r,� 2 � 200 � 400

2)1 �� 59. Verbal Model:

Labels: Let wind speed, then the rate to thecity the rate from the city the distance to thecity kilometers, the distance traveled so far in the returntrip kilometers.

Equation:

Wind speed: kilometers per hour6623

6623

� x

180,000 � 2700x

900,000 � 1500x � 720,000 � 1200x

1500�600 � x� � 1200�600 � x�

1500600 � x

�1200

600 � x

� 1500 � 300 � 1200

� 1500� 600 � x,

� 600 � x,x �

time �distance

rate

60. Verbal Model: (Distance) (rate)(time)

Labels: Distance meters

Rate meters per second

Time

Equation:

Light from the sun travels to the earth in 500 seconds orapproximately 8.33 minutes.

500 � t

1.5 � 1011 � �3.0 � 108�t

� t

� 3.0 � 108

� 1.5 � 1011

�61. Verbal Model:

Equation:

The radio wave travels from Mission Control to the moonin 1.28 seconds.

t � 1.28 seconds

t �3.84 � 108 meters

3.0 � 108 meters per second

time �distance

rate

62. Verbal Model:

Labels: Height of tree height of tree’s shadow meters,height of lamppost meters, height of lamppost’s shadow meter

Equation:

The tree is meters tall.2113

h �8�2�0.75

� 2113

h

8�

2

0.75

� 0.75� 2� 8� h,

�height of tree��height of tree's shadow�

��height of lamppost�

�height of lamppost's shadow�

63. Verbal Model:

Label: Let h height of the building in feet.

Equation:

The Chrysler building is 1044 feet tall.

h � 1044 feet

13

h � 348

h

87�

41�3

h feet

87 feet�

4 feet1�3 foot

�height of building��length of building's shadow�

��height of stake�

�length of stake's shadow�

Page 33: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.3 Modeling with Linear Equations 99

64. (a)

h

30 ft5 ft

6 ft

(b) Verbal Model:

Labels: Height of pole height of pole’s shadow feet,height of person feet, height of person’s shadow feet

Equation:

The pole is 42 feet tall.

h �6

5� 35 � 42

h

35�

6

5

� 5� 6� 30 � 5 � 35� h,

�height of pole��height of pole’s shadow�

��height of person�

�height of person’s shadow�

65. Verbal Model:

Label: Let length of person’s shadow.

Equation:

x � 4.36 feet

44x � 192

50x � 192 � 6x

50x � 6�32 � x�

50

32 � x�

6x

x �

50 ft

6 ft

32 ft x

�height of silo��length of silo’s shadow�

��height of person�

�height of person’s shadow�

66. Verbal Model: (interest from 5%) (total interest)

Labels: Amount invested at amount invested at 5%

interest from interest from total annual interest $580

Equation:

The smallest amount that can be invested at 5% is 12,000 � 4000 � $8000.

x � 4000

�0.005x � �20

0.045x � 600 � 0.05x � 580

0.045x � 0.05�12,000 � x� � 580

�5% � �12,000 � x��0.05�,412% � x�0.045�,

� 12,000 � x,412% � x,

��Interest from 412%� �

67. Verbal Model: interest in (total interest)

Labels: Let x amount in the 3% fund. Then amount in the fund.

Equation:

25,000 � x � $16,666.67 at 412%

x � $8333.33 at 3%

�125 � �0.015x

1000 � 0.03x � 1125 � 0.045x

1000 � 0.03x � 0.045�25,000 � x�41

2%25,000 � x ��

�412% fund��Interest in 3% fund� � �

68. Verbal Model: (Profit from dogwood trees) (profit from red maple trees) (total profit)

Labels: Inventory of dogwood trees inventory of red maple trees

profit from dogwood trees profit from red maple trees

total profit

Equation:

The amount invested in dogwood trees was $7500 and the amount invested in red maple trees was 20,000 � 7500 � $12,500.

x � 7500

0.08x � 600

0.25x � 3400 � 0.17x � 4000

0.25x � 0.17�20,000 � x� � 4000

� 0.20�20,000� � 4000

� 0.17�20,000 � x�,� 0.25x,

� 20,000 � x,� x,

��

Page 34: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

100 Chapter 1 Equations, Inequalities, and Mathematical Modeling

69. Verbal Model: (profit on minivans) (profit on SUVs) (total profit)

Labels: Let x amount invested in minivans. Then, amount invested in SUVs.

Equation:

$450,000 is invested in minivans and is invested in SUVs.600,000 � x � $150,000

x � 450,000

�0.04x � �18,000

0.24x � 168,000 � 0.28x � 150,000

0.24x � 0.28�600,000 � x� � 0.25�600,000�

600,000 � x ��

��

70. (a) Verbal Model:

Labels: Amount in Solution 1 amount in Solution 2amount in final solution

Equation:

Use 25 gallons of Solution 1 and gallons of Solution 2.

(b) Verbal Model:

Labels: Amount in Solution 1 amount in Solution 2amount in final solution

Equation:

Use 4 liters of Solution 1 and liter of Solution 2.

(c) Verbal Model:

Labels: Amount in Solution 1 amount in Solution 2amount in final solution

Equation:

Use 5 quarts of Solution 1 and quarts of Solution 2.

(d) Verbal Model:

Labels: Amount in Solution 1 amount in Solution 2amount in final solution

Equation:

Use 18.75 gallons of Solution 1 and gallons of Solution 2.25 � 18.75 � 6.25

x � 18.75

�0.20x � �3.75

0.70x � 22.5 � 0.90x � 18.75

0.70x � 0.90�25 � x� � 0.75�25�

� 0.75�25�� 0.90�25 � x�,� 0.70x,

�Amount inSolution 1� � �amount in

Solution 2� � � amount infinal solution�

10 � 5 � 5

x � 5

�0.30x � �1.5

0.15x � 4.5 � 0.45x � 3

0.15x � 0.45�10 � x� � 0.30�10�

� 0.30�10�� 0.45�10 � x�,� 0.15x,

�Amount inSolution 1� � �amount in

Solution 2� � � amount infinal solution�

5 � 4 � 1

x � 4

�0.25x � �1

0.25x � 2.5 � 0.50x � 1.5

0.25x � 0.50�5 � x� � 1.5

� 0.30�5�� 0.50�5 � x�,� 0.25x,

�Amount inSolution 1� � �amount in

Solution 2� � � amount infinal solution�

100 � 25 � 75

x � 25

�0.20x � �5

0.10x � 30 � 0.30x � 25

0.10x � 0.30�100 � x� � 0.25�100�

� 0.25�100�� 0.30�100 � x�,� 0.10x,

�Amount inSolution 1� � �amount in

Solution 2� � � amount infinal solution�

Page 35: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.3 Modeling with Linear Equations 101

71. Verbal Model:

Label: Let x amount of 100% concentrate

Equation:

Approximately 32.1 gallons of the 100% concentrate will be needed.

x � 32.1 gallons

41.25 � 0.60x � 22

41.25 � 22 � 0.40x � 1.00x

0.75�55� � 0.40�55 � x� � 1.00x

�Final concentration��Amount� � �Solution 1 concentration��Amount� � �Solution 2 concentration��Amount�

72. Verbal Model:

Labels: Amount of gasoline in Solution 1 amount of gasoline in Solution amount of gasoline in final solution

Equation:

Multiply both sides by 33(41).

Approximately 0.48 gallon of gasoline must be added.

x � 0.48

33x � 16

2624 � 1353x � 2640 � 1320x

6433 � x �

8041 �

4041x

6433 � x �

4041�2 � x�

�4041�2 � x�

2 � x,�3233�2�,

�Amount of gasolinein Solution 1 � � �amount of gasoline

in Solution 2 � � �amount of gasolinein final solution �

73. Verbal Model: (price per pound of peanuts)(number of pounds of peanuts) (price per pound of walnuts)(number ofpounds of walnuts) (price per pound of nut mixture)(number of pounds of nut mixture)

Labels: Let x number of pounds of $2.49 peanuts. Then number of pounds of $3.89 walnuts.

Equation:

Use 50 pounds of peanuts and 50 pounds of walnuts.

100 � x � 50 pounds of $3.89 walnuts

x � 50 pounds of $2.49 peanuts

x ��70

�1.40

�1.40x � �70

2.49x � 389 � 3.89x � 319

2.49x � 3.89�100 � x� � 3.19�100�

100 � x ��

��

74. Verbal Model: (Fixed costs) (variable cost per unit)(number of units) (total costs)

Labels: Fixed costs variable costs total costs

Equation:

The company can produce no more than 8823 units with $85,000 budgeted monthly for costs.

x � 8823.53

8.50x � 75,000

10,000 � 8.50x � 85,000

� $85,000� 8.50x,� $10,000,

��

75. Verbal Model:

Labels:

Equation:

At most the company can manufacture 8064 units.

x �75,000

9.3� 8064.52 units

$85,000 � $10,000 � $9.30x

Total cost � $85,000, variable costs � $9.30x, fixed costs � $10,000

Total cost � �Fixed cost� � �Variable cost per unit� � �Number of units�

Page 36: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

102 Chapter 1 Equations, Inequalities, and Mathematical Modeling

77.

2Ab

� h

2A � bh

A �12

bh

78.

b �2Ah

� a

2Ah

� a � b

2A � �a � b�h

A �12

�a � b�h 79.

S

1 � R� C

S � C�1 � R�

S � C � RC 80.

A � P

Pt� r

A � P � Prt

A � P � Prt

81.

3V

4�a2 � b

�3�4�V

�a2 � b

34

V � �a2b

V �43

�a2b 82.

3V � �h3

3�h2� r

3V � �h3 � 3�rh2

3V � 3�rh2 � �h3

3V � �h2�3r � h�

3V �1

3�h2�3r � h� 83.

2�h � v0t�

t2� a

2�h � v0t� � at2

h � v0t �1

2at2

h � v0t �1

2at2

84.

�R1 is the reciprocal of 1

R1

.�

R1 ��n � 1� f R2

R2 � �n � 1� f

R2 � �n � 1� f

�n � 1� f R2

�1

R1

1

�n � 1� f�

1

R2

�1

R1

1

�n � 1� f�

1

R1

�1

R2

1

f� �n � 1�� 1

R1

�1

R2�

76. Verbal Model: (Volume) (length)(width)(height)

Labels:

length 12 feet, width 3 feet,height

Equation:

The depth of the water is 0.26 foot.

0.26 � h

9.3576 � 36h

9.3576 � �12��3�h

� h��

� 9.3576 cubic feet, Volume � 70 gallons � 0.13368 cubic feet

86.

2S � n�n � 1�d

2n� a

2S � n�n � 1�d

n� 2a

2Sn

� �n � 1�d � 2a

2Sn

� 2a � �n � 1�d

S �n2

�2a � �n � 1�d�85.

CC2

C2 � C� C1

C2 � C

CC2�

1C1

1C

�1

C2�

1C1

1C

�1

C1�

1C2

C �1

1C1

�1

C2

Page 37: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.3 Modeling with Linear Equations 103

91.

r � 34.47�

� 1.12 inches

17.88

4�� r3

17.88 � 4�r3

5.96 �43

�r3

V �43

�r3 92.

h �V

�r2 �603.2� �2�2 � 48 feet

V � �r2h

93.

When

59�64.4 � 32� � 18�C.

F � 64.4�,

C �59�F � 32� 94.

When 59�39.1 � 32� � 3.9�C.

F � 39.1�,

C �59�F � 32� 95.

When

95�50� � 32 � 122�F.

C � 50�,

F �95C � 32

96.

When

95

��30� � 32 � �22�F.

C � �30�,

F �95

C � 32 97. False, it should be written as

z3 � 8z2 � 9

.

98. False

� 860.290 in.3

�43

� �5.9�3

Sphere: V �43

�r3

� 857.375 in.3 � 9.53

Cube: V � s3

99.

(a) If then a and b have the same sign andis negative.

(b) If then a and b have opposite signs andis positive.x � �b�a

ab < 0,

x � �b�aab > 0,

ax � b � 0 ⇒ x � �ba

100. Answers will vary. Sample answer: x � 7 � 4

101. �5x4��25x2��1 �5x4

25x2�

x2

5, x 0 102. 150s2t3 � 25 � 6s2t2t � 5st6t

90.

feet from the person x � 32

3

750x � 2750

200x � 2750 � 550x

200x � 550�5 � x�

L � 5 feet

W2 � 550 pounds

W1 � 200 pounds

W1x � W2�L � x�

89.

x � 6 feet from 50-pound child.

125x � 750

50x � 750 � 75x

50x � 75�10 � x�

W1x � W2�L � x�88.

r �S � a

S � L

r�S � L� � S � a

Sr � rL � S � a

Sr � S � rL � a

S�r � 1� � rL � a

S �rL � a

r � 187.

L � a � d

d� n

L � a � d � nd

L � a � nd � d

L � a � �n � 1�d

Page 38: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

104 Chapter 1 Equations, Inequalities, and Mathematical Modeling

103. 3

x � 5�

2

5 � x�

3

x � 5�

��1��2�x � 5

�3 � 2

x � 5�

1

x � 5

104.

��2x2 � 30x � 45x�x � 3��x � 3�

�5x2 � 45 � 3x2 � 10x2 � 30x

x�x � 3��x � 3�

�5�x2 � 9� � 3x2 � 10x2 � 30x

x�x � 3��x � 3�

5

x�

3x

x2 � 9�

10

x � 3�

5�x � 3��x � 3� � 3x2 � 10x�x � 3�x�x � 3��x � 3�

105.10

7�3�

107�3

��3�3

�10�3

21106. �

4 3�363�216

�4 3�36

6�

2 3�62

3 4

3�6�

43�6

�3�363�36

107.

� �11 � �6

� ���6 � �11 �

�5�6 � 5�11

�5

5

�6 � �11�

5

�6 � �11��6 � �11

�6 � �11108.

�42�10 � 14

89

�14�3�10 � 1�

89

�14�3�10 � 1�

9�10� � 1

�14�3�10 � 1��3�10�2 � �1�2

14

3�10 � 1�

143�10 � 1

�3�10 � 1

3�10 � 1

Section 1.4 Quadratic Equations and Applications

■ You should be able to solve a quadratic equation by factoring, if possible.

■ You should be able to solve a quadratic equation of the form by extracting square roots.

■ You should be able to solve a quadratic equation by completing the square.

■ You should know and be able to use the Quadratic Formula: For

■ You should be able to determine the types of solutions of a quadratic equation by checking the discriminant

(a) If there are two distinct real solutions. The graph has two x-intercepts.

(b) If there is one repeated real solution. The graph has one x-intercept.

(c) If there is no real solution. The graph has no x-intercepts.

■ You should be able to use your calculator to solve quadratic equations.

■ You should be able to solve applications involving quadratic equations. Study the examples in the text carefully.

b2 � 4ac < 0,

b2 � 4ac � 0,

b2 � 4ac > 0,

b2 � 4ac.

x ��b ± �b2 � 4ac

2a.

ax2 � bx � c � 0, a � 0,

u2 � d

Page 39: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.4 Quadratic Equations and Applications 105

1.

General form: 2x2 � 8x � 3 � 0

2x2 � 3 � 8x 2.

General form: x2 � 16x � 0

x2 � 16x 3.

General form: x2 � 6x � 6 � 0

x2 � 6x � 9 � 3

�x � 3�2 � 3

4.

General form:

�3x2 � 42x � 134 � 0

13 � 3x2 � 42x � 147 � 0

13 � 3�x2 � 14x � 49� � 0

13 � 3�x � 7�2 � 0 5.

General form: 3x2 � 90x � 10 � 0

3x2 � 10 � 90x

15�3x2 � 10� � 18x 6.

General form: 4x2 � 2x � 1 � 0

��1���4x2 � 2x � 1� � �1�0�

�4x2 � 2x � 1 � 0

x2 � 2x � 5x2 � 1

x�x � 2� � 5x2 � 1

7.

x � 0 or x � �12

3x � 0 or 2x � 1 � 0

3x�2x � 1� � 0

6x2 � 3x � 0 8.

3x � 1 � 0 ⇒ x �13

3x � 1 � 0 ⇒ x � �13

�3x � 1��3x � 1� � 0

9x2 � 1 � 0 9.

x � 4 or x � �2

x � 4 � 0 or x � 2 � 0

�x � 4��x � 2� � 0

x2 � 2x � 8 � 0

10.

x � 1 � 0 ⇒ x � 1

x � 9 � 0 ⇒ x � 9

�x � 9��x � 1� � 0

x2 � 10x � 9 � 0 11.

x � �5

x � 5 � 0

�x � 5�2 � 0

x2 � 10x � 25 � 0 12.

2x � 3 � 0 ⇒ x � �32

�2x � 3��2x � 3� � 0

4x2 � 12x � 9 � 0

13.

x � 3 or x � �12

3 � x � 0 or 1 � 2x � 0

�3 � x��1 � 2x� � 0

3 � 5x � 2x2 � 0 14.

x � 11 � 0 ⇒ x � 11

2x � 3 � 0 ⇒ x � �32

�2x � 3��x � 11� � 0

2x2 � 19x � 33 � 0

2x2 � 19x � 33 15.

x � �6 or x � 2

x � 6 � 0 or x � 2 � 0

�x � 6��x � 2� � 0

x2 � 4x � 12 � 0

x2 � 4x � 12

16.

x � 2 � 0 ⇒ x � 2

x � 6 � 0 ⇒ x � 6

�x � 6��x � 2� � 0

x2 � 8x � 12 � 0

��1���x2 � 8x � 12� � ��1��0�

�x2 � 8x � 12 � 0

�x2 � 8x � 12 17.

x � �203 or x � �4

3x � 20 � 0 or x � 4 � 0

�3x � 20��x � 4� � 0

3x2 � 32x � 80 � 0

4�34x2 � 8x � 20� � 4�0�

34x2 � 8x � 20 � 0 18.

x � 8 � 0 ⇒ x � �8

x � 16 � 0 ⇒ x � 16

�x � 16��x � 8� � 0

x2 � 8x � 128 � 0

18x2 � x � 16 � 0

Vocabulary Check

1. quadratic equation 2. factoring, extracting square roots, completing the square,and the Quadratic Formula

3. discriminant 4. position equation; velocity of theobject; initial height of the object

5. Pythagorean Theorem

�16t 2 � v0t � s0;

Page 40: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

106 Chapter 1 Equations, Inequalities, and Mathematical Modeling

19.

x � �a

x � a � 0

�x � a�2 � 0

x2 � 2ax � a2 � 0 20.

x � �a � b� � 0 ⇒ x � �a � b

x � �a � b� � 0 ⇒ x � �a � b

�x � �a � b���x � �a � b�� � 0

��x � a� � b���x � a� � b� � 0

�x � a�2 � b2 � 0 21.

x � ±7

x2 � 49

22.

x � ±�169 � ±13

x2 � 169 23.

x � ±�11

x2 � 11 24.

x � ±�32 � ±4�2

x2 � 32

25.

x � ±3�3

x2 � 27

3x2 � 81 26.

x � ±�4 � ±2

x2 � 4

9x2 � 36 27.

x � 16 or x � 8

x � 12 ± 4

x � 12 � ±4

�x � 12�2 � 16

28.

x � �13 ± 5 � �8, �18

x � 13 � ±5

x � 13 � ±�25

�x � 13�2 � 25 29.

x � �2 ± �14

x � 2 � ±�14

�x � 2�2 � 14 30.

x � 5 ± �30

x � 5 � ±�30

�x � 5�2 � 30

31.

x �1 ± 3�2

2

2x � 1 ± 3�2

2x � 1 � ±�18

�2x � 1�2 � 18 32.

x ��7 ± 2�11

4� �

74

±�11

2

4x � �7 ± 2�11

4x � 7 � ±�44

�4x � 7�2 � 44

33.

The only solution to the equation is x � 2.

x � 2

�7 � 3 or 2x � 4

x � 7 � x � 3 or x � 7 � �x � 3

x � 7 � ± �x � 3�

�x � 7�2 � �x � 3�2 34.

or

or

The only solution to the equation is x � �92.

x � �92

2x � �9

x � 5 � �x � 4 5 � 4

x � 5 � ��x � 4�x � 5 � ��x � 4�

x � 5 � ± �x � 4�

�x � 5�2 � �x � 4�2

35.

x � 4 or x � �8

x � �2 ± 6

x � 2 � ±6

�x � 2�2 � 36

x2 � 4x � 22 � 32 � 22

x2 � 4x � 32

x2 � 4x � 32 � 0 36.

x � 3 or x � �1

x � 1 ± 2

x � 1 � ±�4

�x � 1�2 � 4

x2 � 2x � ��1�2 � 3 � ��1�2

x2 � 2x � 3

x2 � 2x � 3 � 0 37.

x � �6 ± �11

x � 6 � ±�11

�x � 6�2 � 11

x2 � 12x � 62 � �25 � 62

x2 � 12x � �25

x2 � 12x � 25 � 0

Page 41: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.4 Quadratic Equations and Applications 107

38.

x � �4 ± �2

x � 4 � ±�2

�x � 4�2 � 2

x2 � 8x � 42 � �14 � 16

x2 � 8x � �14

x2 � 8x � 14 � 0 39.

x � 1 ±�63

x � 1 ± �23

x � 1 � ±�23

�x � 1�2 �23

x2 � 2x � 12 � �13

� 12

x2 � 2x � �13

9x2 � 18x � �3 40.

x �2

3± �2

x �2

3� ±�2

�x �2

3�2

� 2

�x �2

3�2

�18

9

x2 �4

3x � ��

2

3�2

�14

9�

4

9

x2 �4

3x �

14

9

9x2 � 12x � 14

41.

x � 2 ± 2�3

x � 2 � ±�12

�x � 2�2 � 12

x2 � 4x � 22 � 8 � 22

x2 � 4x � 8

x2 � 4x � 8 � 0

�x2 � 4x � 8 � 0

8 � 4x � x2 � 0 42.

No real solution

�x �12�2

� �34

x2 � x �14 � �1 �

14

x2 � x � 1 � 0

�x2 � x � 1 � 0

45.

�1

�x � 1�2 � 4

1

x2 � 2x � 5�

1x2 � 2x � 12 � 12 � 5

46.

�1

�x � 6�2 � 17

1

x2 � 12x � 19�

1�x2 � 12x � ��6�2� � 19 � 36

43.

x ��5 ± �89

4

x � �54

±�89

4

x �54

� ±�89

4

�x �54�

2

�8916

x2 �52

x � �54�

2

� 4 � �54�

2

x2 �52

x � 4

2x2 � 5x � 8

2x2 � 5x � 8 � 0 44.

x �1

2± 5 �

11

2, �

9

2

x �1

2� ±�25

�x �1

2�2

� 25

�x �1

2�2

�100

4

x2 � x � ��1

2�2

�99

4�

1

4

x2 � x �99

4

4x2 � 4x � 99 � 0

Page 42: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

108 Chapter 1 Equations, Inequalities, and Mathematical Modeling

47.

�4

�x � 2�2 � 7

4

x2 � 4x � 3�

4x2 � 4x � 4 � 4 � 3

48.

�5

�x �252 �2

� 5814

5

x2 � 25x � 11�

5

x2 � 25x � �252 �2

� 11 �6254

49.

�1

���x � 3�2 � 9�

1�9 � �x � 3�2

�1

��1��x � 3�2 � 9�

1

�6x � x2�

1��1�x2 � 6x � 32 � 32�

50.

�1

�25 � �x � 3�2

�1

�16 � �x2 � 6x � 32� � 9

1

�16 � 6x � x2�

1

�16 � 1�x2 � 6x�

51. (a)

(b) The x-intercepts are ��1, 0� and ��5, 0�.

−10 5

−6

4

y � �x � 3�2 � 4 (c)

(d) The x-intercepts of the graph are solutions to theequation 0 � �x � 3�2 � 4.

x � �1 or x � �5

�3 ± 2 � x

±�4 � x � 3

4 � �x � 3�2

0 � �x � 3�2 � 4

52. (a)

(b) The x-intercepts are �5, 0� and �3, 0�.

−1 8

2

4

y � �x � 4�2 � 1 (c)

(d) The x-intercepts of the graph are solutions to the equation0 � �x � 4�2 � 1.

x � 4 ± 1 � 5, 3

x � 4 � ±�1

�x � 4�2 � 1

0 � �x � 4�2 � 1

53. (a)

(b) The x-intercepts are �1, 0� and �3, 0�.

−3

−4

6

2

y � 1 � �x � 2�2 (c)

(d) The x-intercepts of the graph are solutions to the equation 0 � 1 � �x � 2�2.

x � 3 or x � 1

x � 2 ± 1

x � 2 � ±1

�x � 2�2 � 1

0 � 1 � �x � 2�2

54. (a)

(b) The x-intercepts are �5, 0� and �11, 0�.

0 18

−2

10

y � 9 � �x � 8�2 (c)

(d) The x-intercepts of the graph are solutions to the equation0 � 9 � �x � 8�2.

x � 8 ± 3 � 11, 5

x � 8 � ±�9

�x � 8�2 � 9

0 � 9 � �x � 8�2

Page 43: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.4 Quadratic Equations and Applications 109

55. (a)

(b) The x-intercepts are and

(d) The x-intercepts of the graph are solutions to theequation 0 � �4x2 � 4x � 3.

�32, 0�.��1

2, 0�

−5 7

−3

5

y � �4x2 � 4x � 3 (c)

x �32 or x � �

12

x �12 ± 1

x �12 � ±�1

�x �12�2

� 1

x2 � x � �12�2 �

34 � �1

2�2

x2 � x �34

4�x2 � x� � 3

4x2 � 4x � 3

0 � �4x2 � 4x � 3

56. (a)

(b) The x-intercepts are and .�12, 0���1

2, 0�

−3 3

−2

2

y � 4x2 � 1 (c)

(d) The x-intercepts of the graph are solutions to the equation0 � 4x2 � 1.

x � ±�14 � ±1

2

x2 �14

4x2 � 1

0 � 4x2 � 1

57. (a)

(b) The x-intercepts are and �1, 0�.��4, 0�

−8 4

−7

1

y � x2 � 3x � 4 (c)

(d) The x-intercepts of the graph are solutions to the equation 0 � x2 � 3x � 4.

x � �4 or x � 1

x � 4 � 0 or x � 1 � 0

0 � �x � 4��x � 1�

0 � x2 � 3x � 4

58. (a)

(b) The x-intercepts are and ��3, 0�.�8, 0�

−5 10

−35

5

y � x2 � 5x � 24 (c)

(d) The x-intercepts of the graph are solutions to the equation0 � x2 � 5x � 24.

x � 3 � 0 ⇒ x � �3

x � 8 � 0 ⇒ x � 8

�x � 8��x � 3� � 0

0 � x2 � 5x � 24

59.

No real solution

b2 � 4ac � ��5�2 � 4�2��5� � �15 < 0

2x2 � 5x � 5 � 0 60.

Two real solutions

� 16 � 20 � 36 > 0

b2 � 4ac � ��4�2 � 4��5��1�

�5x2 � 4x � 1 � 0

61.

Two real solutions

b2 � 4ac � ��1�2 � 4�2���1� � 9 > 0

2x2 � x � 1 � 0 62.

One repeated solution

� 16 � 16 � 0

b2 � 4ac � ��4�2 � 4�1��4�

x2 � 4x � 4 � 0

Page 44: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

110 Chapter 1 Equations, Inequalities, and Mathematical Modeling

63.

No real solution

b2 � 4ac � ��5�2 � 4�13��25� � �

253 < 0

13x2 � 5x � 25 � 0 64.

One repeated solution

� 64 � 64 � 0

b2 � 4ac � ��8�2 � 4�47 ��28�

47x2 � 8x � 28 � 0

65.

Two real solutions

b2 � 4ac � �1.2�2 � 4�0.2���8� � 7.84 > 0

0.2x2 � 1.2x � 8 � 0 66.

Two real solutions

� 5.76 � 298.8 � 304.56 > 0

b2 � 4ac � �2.4�2 � 4��8.3��9�

9 � 2.4x � 8.3x2 � 0

67.

��1 ± 3

4�

12

, �1

��1 ± �12 � 4�2���1�

2�2�

x ��b ± �b2 � 4ac

2a

2x2 � x � 1 � 0 68.

�1 ± 3

4� 1, �

1

2

�1 ± �1 � 8

4

����1� ± ���1�2 � 4�2���1�

2�2�

x ��b ± �b2 � 4ac

2a

2x2 � x � 1 � 0 69.

��8 ± 16

32�

14

, �34

��8 ± �82 � 4�16���3�

2�16�

x ��b ± �b2 � 4ac

2a

16x2 � 8x � 3 � 0

70.

�20 ± 10

50�

3

5,

1

5

�20 ± �400 � 300

50

����20� ± ���20�2 � 4�25��3�

2�25�

x ��b ± �b2 � 4ac

2a

25x2 � 20x � 3 � 0 71.

��2 ± 2�3

�2� 1 ± �3

��2 ± �22 � 4��1��2�

2��1�

x ��b ± �b2 � 4ac

2a

�x2 � 2x � 2 � 0

2 � 2x � x2 � 0

73.

��14 ± 2�5

2� �7 ± �5

��14 ± �142 � 4�1��44)

2�1�

x ��b ± �b2 � 4ac

2a

x2 � 14x � 44 � 0 74.

� �3 ± �13

��6 ± 2�13

2

��6 ± �36 � 16

2

��6 ± �62 � 4�1���4�

2�1�

x ��b ± �b2 � 4ac

2a

x2 � 6x � 4 � 0

6x � 4 � x2 75.

��8 ± 4�5

2� �4 ± 2�5

��8 ± �82 � 4�1���4�

2�1�

x ��b ± �b2 � 4ac

2a

x2 � 8x � 4 � 0

72.

�10 ± 2�3

2� 5 ± �3

�10 ± �100 � 88

2

����10� ± ���10�2 � 4�1��22�

2�1�

x ��b ± �b2 � 4ac

2a

x2 � 10x � 22 � 0

Page 45: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.4 Quadratic Equations and Applications 111

76.

�1

2±�5

2

�1 ± �1 � 4

2

����1� ± ���1�2 � 4�1���1�

2�1�

x ��b ± �b2 � 4ac

2a

x2 � x � 1 � 0

4x2 � 4x � 4 � 0 77.

��12 ± 6�7

�18�

23

±�73

��12 ± �122 � 4��9��3�

2��9�

x ��b ± �b2 � 4ac

2a

�9x2 � 12x � 3 � 0

12x � 9x2 � �3 78.

�5

4±�3

4

�20 ± �400 � 352

16

����20� ± ���20�2 � 4�8��11�

2�8�

x ��b ± �b2 � 4ac

2a

8x2 � 20x � 11 � 0

16x2 � 22 � 40x

79.

� �43

��24 ± 0

18

��24 ± �242 � 4�9��16�

2�9�

x ��b ± �b2 � 4ac

2a

9x2 � 24x � 16 � 0 80.

� �1

3±�11

6

��24 ± ��144��11�

72

��24 ± �576 � 1008

72

��24 ± �242 � 4�36���7�

2�36�

x ��b ± �b2 � 4ac

2a

36x2 � 24x � 7 � 0 81.

��4 ± 8�2

8� �

12

± �2

��4 ± �42 � 4�4���7�

2�4�

x ��b ± �b2 � 4ac

2a

4x2 � 4x � 7 � 0

4x2 � 4x � 7

82.

�5

4±�5

2

�40 ± 16�5

32

�40 ± �1600 � 320

32

����40� ± ���40�2 � 4�16��5�

2�16�

x ��b ± �b2 � 4ac

2a

16x2 � 40x � 5 � 0 83.

��28 ± 0

�98�

27

��28 ± �282 � 4��49���4�

2��49�

x ��b ± �b2 � 4ac

2a

�49x2 � 28x � 4 � 0

28x � 49x2 � 4 84.

��3 ± �13

2� �

3

2±�13

2

��3 ± �32 � 4�1���1�

2�1�

x ��b ± �b2 � 4ac

2a

x2 � 3x � 1 � 0

3x � x2 � 1 � 0

85.

��8 ± 2�6

�4� 2 ±

�62

��8 ± �82 � 4��2���5�

2��2�

t ��b ± �b2 � 4ac

2a

�2t2 � 8t � 5 � 0

8t � 5 � 2t2 86.

� �8

5±�3

5

� �8

10�3

50

��80 ± �6400 � 6100

50

��80 ± �802 � 4�25��61�

2�25�

h ��b ± �b2 � 4ac

2a

25h2 � 80h � 61 � 0 87.

�12 ± 2�11

2� 6 ± �11

����12� ± ���12�2 � 4�1��25�

2�1�

y ��b ± �b2 � 4ac

2a

y2 � 12y � 25 � 0

�y � 5�2 � 2y

Page 46: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

112 Chapter 1 Equations, Inequalities, and Mathematical Modeling

88.

� �7 ± �13

��14 ± �52

2

��14 ± �142 � 4�1��36�

2�1�

z ��b ± �b2 � 4ac

2a

z2 � 14z � 36 � 0

z2 � 12z � 36 � �2z

�z � 6�2 � �2z 89.

� �38

±�265

8

��3 ± �265

8

��3 ± �32 � 4�4���16�

2�4�

x ��b ± �b2 � 4ac

2a

4x2 � 3x � 16 � 0

4x2 � 3x � 16

12

x2 �38

x � 2

90.

�686 ± 196�6

25

�1372 ± �921,984

50

����1372� ± ���1372�2 � 4�25��9604�

2�25�

x ��b ± �b2 � 4ac

2a

25x2 � 1372x � 9604 � 0

2549

x2 � 28x � 196 � 0

2549

x2 � 20x � 196 � 8x

�57

x � 14�2

� 8x 91.

x � 0.976, �0.643

x �1.7 ± ���1.7�2 � 4�5.1���3.2�

2�5.1�

5.1x2 � 1.7x � 3.2 � 0

92.

� 1.400, �0.150

�2.50 ± �9.61

4

����2.50� ± ���2.50�2 � 4�2���0.42�

2�2�

x ��b ± �b2 � 4ac

2a

2x2 � 2.50x � 0.42 � 0 93.

x � �14.071, 1.355

x ����0.852� ± ���0.852�2 � 4��0.067��1.277�

2��0.067�

�0.067x2 � 0.852x � 1.277 � 0

94.

� 2.137, 18.063

��0.101 ± �0.006341

�0.01

��0.101 ± ��0.101�2 � 4��0.005���0.193�

2��0.005�

x ��b ± �b2 � 4ac

2a

�0.005x2 � 0.101x � 0.193 � 0 95.

x � 1.687, �0.488

x �506 ± ���506�2 � 4�422���347�

2�422�

422x2 � 506x � 347 � 0

Page 47: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.4 Quadratic Equations and Applications 113

96.

� 0.672, �0.968 ��326 ± �3,252,276

2200

��326 ± ��326�2 � 4�1100���715�

2�1100�

x ��b ± �b2 � 4ac

2a

1100x2 � 326x � 715 � 0 97.

x � �2.200, �0.290

x ��31.55 ± ��31.55�2 � 4�12.67��8.09�

2�12.67�

12.67x2 � 31.55x � 8.09 � 0

98.

� �2.995, 2.971 �0.08 ± �369.031

�6.44

����0.08� ± ���0.08�2 � 4��3.22��28.651�

2��3.22�

x ��b ± �b2 � 4ac

2a

�3.22x2 � 0.08x � 28.651 � 0 99. Complete the square.

x � 1 ± �2

x � 1 � ± �2

�x � 1�2 � 2

x2 � 2x � 12 � 1 � 12

x2 � 2x � 1

x2 � 2x � 1 � 0

103. Complete the square.

x �12 ± �3

x �12 � ±�12

4

�x �12�2

�124

x2 � x � �12�2

�114 � �1

2�2

x2 � x �114

x2 � x �114 � 0

104. Complete the square.

x � �32 ± �3

x �32 � ±�3

�x �32�2

� 3

x2 � 3x � �32�2

�34 �

94

x2 � 3x �34 � 0 105. Extract square roots.

x � �12

2x � �1

For x � ��x � 1�:

0 � 1 No solution

For x � ��x � 1�:

x � ± �x � 1�

x2 � �x � 1�2

�x � 1�2 � x2

106. Factor.

ax � b � 0 ⇒ x �b

a

ax � b � 0 ⇒ x � �b

a

�ax � b��ax � b� � 0

a2x2 � b2 � 0

100. Factor.

x � �3

x � 0 or x � 3 � 0

x�x � 3� � 0

11�x2 � 3x� � 0

11x2 � 33x � 0 101.

x � 6 or x � �12

x � 3 � 9 or x � 3 � �9

x � 3 � ±9

�x � 3�2 � 81 Extract square roots.

102. Extract square roots.

x � 7

x � 7 � 0

�x � 7�2 � 0

x2 � 14x � 49 � 0

Page 48: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

114 Chapter 1 Equations, Inequalities, and Mathematical Modeling

107. Quadratic Formula

�34

±�97

4

�3 ± �97

4

x ����3� ± ���3�2 � 4�2���11�

2�2�

0 � 2x2 � 3x � 11

3x � 4 � 2x2 � 7 108. Factor.

x � 1 x � �1

x � 1 � 0 or x � 1 � 0

�x � 1��x � 1� � 0

4�x2 � 1� � 0

4x2 � 4 � 0

4x2 � 2x � 4 � 2x � 8

109. (a)

(b)

(c)

Since w must be greater than zero, we have feet and the length is w � 14 � 48 feet.

w � 34

w � �48 or w � 34

�w � 48��w � 34� � 0

w2 � 14w � 1632 � 0

w�w � 14� � 1632

w

w + 14

110. Total fencing:

Total area:

For

There are two solutions: meters and meters or meters and meters.y � 10x � 17.5y �

703

x � 7.5

x � 7.5

2x�703 � � 350

y �703

:

�y � 10 � 0 ⇒ y � 10

3y � 70 � 0 ⇒ y �703

�3y � 70���y � 10� � 0

�3y2 � 100y � 700 � 0

100y � 3y2 � 700 � 0

12

�100y � 3y2� � 350 � 0

2�100 � 3y4 �y � 350

x �100 � 3y

4

2xy � 350

4x � 3y � 100

For

x � 17.5

2x�10� � 350

y � 10:

111.

Since x must be positive, we have inches. Thedimensions of the box are 6 inches � 6 inches � 2 inches.

x � 6

x � �14 or x � 6

0 � �x � 14��x � 6�

0 � x2 � 8x � 84

84 � x2 � 4x�2�

S � x2 � 4xh 112. Volume:

Original size of material:

Side: centimeters

Dimensions: 14 centimeters 14 centimeters�

x � 4 � 10 � 4 � 14

x � ±10

x2 � 100

2x2 � 200

113.

Thus,

—CONTINUED—

a � 1, b � �150, and c � 2500.

4�x2 � 150x � 2500� � 0

4x2 � 600x � 10,000 � 0

20,000 � 600x � 4x2 � 10,000100 ft

200 ft

100 2− x

200 2− x

x

x

�200 � 2x��100 � 2x� �1

2�100��200�

Page 49: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.4 Quadratic Equations and Applications 115

113. —CONTINUED—

not possible since the lot is only 100 feet wide.

The person must go around the lot 19.098 feet

24 inches�

19.098 feet

2 feet� 9.5 times.

x �150 � 111.8034

2� 19.098 feet

x �150 � 111.8034

2� 130.902 feet,

x �150 ± ���150�2 � 4�1��2500�

2�1��

150 ± 111.8034

2

114. Original arrangement: x rows, y seats per row,

New arrangement: rows, seats per row

Originally, there were 8 rows of seats with seats per row.72

8 � 9

x � 6 � 0 ⇒ x � �6

x � 8 � 0 ⇒ x � 8

�x � 8��x � 6� � 0

x2 � 2x � 48 � 0

3x2 � 6x � 144 � 0

72x � 3x2 � 144 � 6x � 72x

�x � 2��72 � 3x� � 72x

x�x � 2��72x

� 3� � 72x

�x � 2��72x

� 3� � 72

�x � 2��y � 3� � 72

�y � 3��x � 2�

y �72x

xy � 72, 115.

(a)

(b) Model:

Labels: Rate 500 miles per hour

Distance

Equation:

The bomb will travel approximately 6.2 miles horizontally.

d � 500�0.0124� � 6.2 miles

� d

Time �20�53600

� 0.0124 hour

�Rate� � �time� � �distance�

� 44.72 seconds

� 20�5 seconds

t � �2000

t2 � 2000

�16t2 � 32,000 � 0

s � �16t2 � 32,000

116. (a)

Since the object was dropped, and the initialheight is Thus,

(b) feet

(c)

It will take the coin about 7.84 seconds to strike the ground.

t ��98416

��246

2� 7.84

t2 �98416

16t2 � 984

0 � �16t2 � 984

s � �16�4�2 � 984 � 728

s � �16t2 � 984.s0 � 984.v0 � 0,

s � �16t2 � v0t � s0 117.

(a)

(b) When

When

When

During the interval the baseball’s speeddecreased due to gravity.

(c) The ball hits the ground when

By the Quadratic Formula, or Assuming that the ball is not caught and drops to theground, it will be in the air for approximately 9.209seconds.

t � 9.209.t � �0.042

�16t 2 � 146 23t � 6 1

4 � 0

s � 0.

3 ≤ t ≤ 5,

s�5� � 339.58 feett � 5:

s�4� � 336.92 feett � 4:

s�3� � 302.25 feett � 3:

s � �16t 2 � 14623t � 61

4

s0 � 614 feet

v0 � 100 mph ��100��5280�

3600� 1462

3 ft�sec

s � �16t 2 � v0t � s0

Page 50: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

116 Chapter 1 Equations, Inequalities, and Mathematical Modeling

118. (a)

Since the object was dropped, and the initial height is Thus,

(b)

(c) The object reaches the ground between seconds and seconds. Numerical approximations will vary, though 10.7 seconds is a reasonable estimate.

(d)

It will take the object about 10.65 seconds to reach the ground.

(e)

The zero of the graph is at seconds.t � 10.65

00 12

2000

t ��181516

�11�15

4� 10.65

t2 �181516

16t2 � 1815

0 � �16t2 � 1815

t � 12t � 10

s � �16t2 � 1815.s0 � 1815.v0 � 0,

s � �16t2 � v0t � s0

Time, t 0 2 4 6 8 10 12

Height, s 1815 1751 1559 1239 791 215 �489

119.

(a)

The average admission price reached or surpassed $5.00 in 1999.

(b)

By the Quadratic Formula, or 42.80. Since we choose which corresponds to 1999.

(c) For 2008, let Answers will vary.P�18� � $6.87.t � 18:

t � 8.68 � 97 ≤ t ≤ 13,t � 8.68

�0.0081t 2 � 0.417t � 3.01 � 0

�0.0081t 2 � 0.417t � 1.99 � 5.00

P � �0.0081t 2 � 0.417t � 1.99, 7 ≤ t ≤ 13

7 8 9 10 11 12 13

$4.51 $4.81 $5.09 $5.35 $5.60 $5.83 $6.04P

t

120. (a)

The median income reached and surpassed $40,000in 1998, x � 8.65.

5 1330,000

45,000 (b) Substituting into the model gives $40,003.46.Substituting into the model gives $39,940.31.

(c) For 2008, and For 2013, and No. The model shows the median income decreasingafter 2002.

I � $30,609.28.t � 23I � $40,108.68.t � 18

x � 8.6x � 8.65

Page 51: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.4 Quadratic Equations and Applications 117

121. Pythagorean Theorem

Each leg in the right triangle is approximately 3.54 centimeters.

�5�2

2� 3.54 centimeters

�5

�2

x ��25

2

x2 �25

2x

x

5

2x2 � 25

x2 � x2 � 52

122. Model:

Labels:

Equation:

Each side of the equilateral triangle is approximately inches long.11.55

s ��4003

�20�3

3� 11.55 inches

s2 �4003

34

s2 � 100

100 �s2

4� s2

102 � �s2�

2� s2

half of side �s2

side � s,height � 10 inches, s

s

10

12

�height�2 � �half of side�2 � �side�2

123.

Using the positive value for we have one plane moving northbound at miles per hour and one plane moving eastbound at miles per hour.r � 550

r � 50 � 600r,

r ��900 ± �9002 � 4�18���5,931,100�

2�18��

�900 ± 60�118,847

36

18r2 � 900r � 5,931,100 � 0

9�r � 50�2 � 9r2 � 24402

dN2 � dE

2 � 24402

dE � �3 hours��r mph�

2440 miles

3r

3(r + 50)

dN � �3 hours��r � 50 mph�

124. (a) Model:

Labels:

Equation:

(b) When

The boat is approximately feet from the dock when there is 75 feet of rope out.73.5

x � �5400 � 30�6 � 73.5 feet

x2 � 5625 � 225 � 5400

152 � x2 � 752l � 75:

152 � x2 � l2

length of rope � ldistance to dock � x,winch � 15,

�winch�2 � �distance to dock�2 � �length of rope�2

Page 52: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

118 Chapter 1 Equations, Inequalities, and Mathematical Modeling

125.

x � 50,000 units

0 � �x � 50,000�2

0 � x2 � 100,000x � 2,500,000,000

0 � 0.0002x2 � 20x � 500,000

x�20 � 0.0002x� � 500,000

126.

x �150,000 ± �150,0002 � 4�550,000,000�

2� 3761 units or 146,239 units

x2 � 150,000x � 550,000,000 � 0

�4x2 � 600,000x � 2,200,000,000 � 0

60x � 0.0004x2 � 220,000 � 0

x�60 � 0.0004x� � 220,000

127.

Using the positive value for x, we have

x ��20 � �7150

0.25� 258 units.

x ��20 ± �202 � 4�0.125���13,500�

2�0.125�

0.125x2 � 20x � 13,500 � 0

0.125x2 � 20x � 500 � 14,000 128.

Choosing the positive value for x, 100 units can beproduced for a cost of $11,500.

x � 100, �130

x ��15 ± �152 � 4(0.5���6500�

2�0.5�

0 � 0.5x2 � 15x � 6500

11,500 � 0.5x2 � 15x � 5000

129.

Choosing the positive value for x, we have

x ��0.04 � �7.0416

0.004� 653 units.

��0.04 ± �7.0416

0.004

x ��0.04 ± ��0.04�2 � 4�0.002���880�

2�0.002�

0.002x2 � 0.04x � 880 � 0

800 � 0.04x � 0.002x2 � 1680 130.

Choosing the positive value for x, 48 units canbe produced for a cost of $896.

x � 48, �8

x ����10� ± ���10�2 � 4�1

4���96�2�1

4�

0 � �96 � 10x �x2

4

896 � 800 � 10x �x2

4

131.

(a)

The total money in circulation reached or surpassed $600 billion in 2001.

(b)

By the Quadratic Formula, or Since we choose which corresponds to 2001.

(c) For 2008, let billionAnswers will vary.

M�18� � $991.98t � 18:

t � 11.15 ≤ t ≤ 13,�13.1.t � 11.1

1.835t 2 � 3.58t � 267.0 � 0

1.835t 2 � 3.58t � 333.0 � 600

M � 1.835t 2 � 3.58t � 333.0, 5 ≤ t ≤ 13

5 6 7 8 9 10 11 12 13

(in billions) $396.78 $420.54 $447.98 $479.08 $513.86 $552.30 $594.42 $640.20 $689.66M

t

Page 53: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.4 Quadratic Equations and Applications 119

132. (a)

Because choose 16.8�C.10 ≤ x ≤ 25,

� �13.1, 16.8

x �1.65 ± ���1.65�2 � 4�0.45���99.25�

2�0.45�

0 � 0.45x2 � 1.65x � 99.25

150 � 0.45x2 � 1.65x � 50.75 (b)

Oxygen consumption is increased by a factor of approximately 2.5.

197.75 � 79.25 � 2.5

x � 20: 0.45�20�2 � 1.65�20� � 50.75 � 197.75

x � 10: 0.45�10�2 � 1.65�10� � 50.75 � 79.25

133.

The other two distances are 354.5 miles and 433.5 miles.

x �1576 ± �15762 � 4�2��307,344�

2�2� � 354.5 or 433.5

0 � 2x2 � 1576x � 307,344

313,600 � x2 � 620,944 � 1576x � x2

5602 � x2 � (788 � x�2

Distance from New Orleans to Austin � 1348 � 560 � x � 788 � x

Distance from Oklahoma City to Austin � x

Distance from Oklahoma City to New Orleans � 560

�Distance from OklahomaCity to New Orleans �

2

� �Distance from OklahomaCity to Austin �

2

� � Distance from New Orleans to Austin�

2

134. (a)

so:

x � 1 � 16.5078 ft

x � 15.5078 ft

4x2 � 4x � 1024 � 0

4x�x � 1� � 1024

V � 1024 ft3,

4

x + 1

x

(b) weighs approximately 62.4 lbs., so

(c)

At 5 gallons per minute:

7660.08 gal5 galsmin

� 1532.02 min � 25.5 hours

1024 ft3

0.13368 ft3�gal� 7660.08 gal

1024 ft3 � 62.4 lbs�ft3 � 63,897.6 lbs.

1 ft3

135. False

so, the quadratic equation has no real solutions.

b2 � 4ac � ��1�2 � 4��3���10� < 0,

136. False. The product must equal zero for the Zero-Factorproperty to be used.

137. The student should have subtracted 15x from both sidesso that the equation is equal to zero. By factoring out anx, there are two solutions, and x � 6.x � 0

138. (a) Neither. A linear equation has at most one real solution, and a quadratic equation has at most two real solutions.

(b) Both

(c) Quadratic

(d) Neither

Page 54: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

120 Chapter 1 Equations, Inequalities, and Mathematical Modeling

139.

(a) Let

or

or x � �5 x � �103

x � 4 � �1 x � 4 �23

u � �1 u �23

u � 1 � 0 3u � 2 � 0

�3u � 2��u � 1� � 0

3u2 � u � 2 � 0

u � x � 4

3�x � 4�2 � �x � 4� � 2 � 0

(b)

or

(c) The method of part (a) reduces the number of algebraicsteps.

x � �5x � �103

x � 5 � 03x � 10 � 0

�3x � 10��x � 5� � 0

3x2 � 25x � 50 � 0

3x2 � 24x � 48 � x � 4 � 2 � 0

3�x2 � 8x � 16� � �x � 4� � 2 � 0

140. (a)

ax � b � 0 ⇒ x � �b

a

x � 0

x�ax � b� � 0

ax2 � bx � 0 (b)

x � 1 � 0 ⇒ x � 1

ax � 0 ⇒ x � 0

ax�x � 1� � 0

ax2 � ax � 0

144. is a factor.

is a factor.

30x2 � 7x � 2 � 0

�6x � 1��5x � 2� � 0

x � �25 ⇒ 5x � �2 ⇒ 5x � 2

x �16 ⇒ 6x � 1 ⇒ 6x � 1

145. and

One possible equation is:

Any non-zero multiple of this equation would also havethese solutions.

x2 � 2x � 1 � 0

x2 � 2x � 1 � 2 � 0

�x � 1�2 � ��2 �2� 0

��x � 1� � �2 ��x � 1� � �2 � 0

�x � �1 � �2� �x � �1 � �2 � � 0

1 � �21 � �2

141.

One possible equation is:

Any non-zero multiple of this equation would also havethese solutions.

x2 � 3x � 18 � 0

�x � 3��x � 6� � 0

�x � ��3���x � 6� � 0

�3 and 6 142.

x2 � 15x � 44 � 0

�x � 4��x � 11� � 0

�x � ��4���x � ��11�� � 0

143. 8 and 14

One possible equation is:

Any non-zero multiple of this equation would also havethese solutions.

x2 � 22x � 112 � 0

�x � 8��x � 14� � 0

146. so:

x2 � 6x � 4 � 0

�x � 3 � �5 ��x � 3 � �5 � � 0

�x � ��3 � �5 ���x � ��3 � �5 �� � 0

x � �3 � �5, x � �3 � �5,

147. by the Associative Property ofMultiplication.�10x�y � 10�xy� 148. by the Distributive Property.�4�x � 3� � �4x � 12

Page 55: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.5 Complex Numbers 121

149. by the Additive Inverse Property.7x 4 � ��7x 4� � 0

159. Answers will vary.

150. by the AssociativeProperty of Addition.�x � 4� � x3 � x � �4 � x3�

151.

� x2 � 3x � 18

�x � 3��x � 6� � x2 � 6x � 3x � 18 152.

� x2 � 9x � 8

�x � 8��x � 1� � x2 � x � 8x � ��8���1�

153.

� x3 � 3x2 � 2x � 8

� x3 � x2 � 2x � 4x2 � 4x � 8

�x � 4��x2 � x � 2� � x�x2 � x � 2� � 4�x2 � x � 2� 154.

� x3 � 3x2 � 50x � 36

�x � 9��x2 � 6x � 4� � x3 � 6x2 � 4x � 9x2 � 54x � 36

155.

� x2�x � 3��x2 � 3x � 9�

� x2�x3 � 33�

x5 � 27x2 � x2�x3 � 27� 156.

� x�x � 7��x � 2�

x3 � 5x2 � 14x � x�x2 � 5x � 14�

157.

� �x � 5��x2 � 2�

x3 � 5x2 � 2x � 10 � x2�x � 5� � 2�x � 5� 158.

� �x � 2��x � 2��2x � 1�

� �x2 � 4��2x � 1�

� x2�2x � 1� � 4�2x � 1�

2x3 � x2 � 8x � 4 � �2x3 � x2� � �8x � 4�

Section 1.5 Complex Numbers

■ Standard form: .

If then is a real number.

If and then is a pure imaginary number.

■ Equality of Complex Numbers: if and only if and

■ Operations on complex numbers

(a) Addition:

(b) Subtraction:

(c) Multiplication:

(d) Division:

■ The complex conjugate of

■ The additive inverse of

■ ��a � �a i for a > 0.

a � bi is �a � bi.

�a � bi��a � bi� � a2 � b2

a � bi is a � bi:

a � bi

c � di�

a � bi

c � di�

c � di

c � di�

ac � bd

c2 � d 2�

bc � ad

c2 � d 2i

�a � bi��c � di� � �ac � bd� � �ad � bc�i

�a � bi� � �c � di� � �a � c� � �b � d�i

�a � bi� � �c � di� � �a � c� � �b � d�i

b � da � ca � bi � c � di

a � bib � 0,a � 0

a � bib � 0,

a � bi

Vocabulary Check

1. (a) iii (b) i (c) ii 2.

3. principal square 4. complex conjugates

��1; �1

Page 56: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

122 Chapter 1 Equations, Inequalities, and Mathematical Modeling

1.

b � 6

a � �10

a � bi � �10 � 6i 2.

b � 4

a � 13

a � bi � 13 � 4i 3.

b � 3 � 8 ⇒ b � 5

a � 1 � 5 ⇒ a � 6

�a � 1� � �b � 3�i � 5 � 8i

4.

a � 0

a � 6 � 6

b � �52

2b � �5

�a � 6� � 2bi � 6 � 5i 5. 4 � ��9 � 4 � 3i 6. 3 � ��16 � 3 � 4i

7.

� 2 � 3�3 i

2 � ��27 � 2 � �27 i 8. 1 � ��8 � 1 � 2�2i 9. ��75 � �75 i � 5�3 i

10. ��4 � 2i 11. 8 � 8 � 0i � 8 12. 45

13.

� �1 � 6i

�6i � i2 � �6i � 1 14.

� 4 � 2i

�4i2 � 2i � �4��1� � 2i 15.

� 0.3i

��0.09 � �0.09 i

16. ��0.0004 � 0.02i 17. �5 � i� � �6 � 2i� � 11� i 18. �13 � 2i�� ��5 � 6i� � 8 �4i

19.

� 4

�8 � i� � �4 � i� � 8 � i � 4 � i 20.

� �3 � 11i

�3 � 2i� � �6 � 13i� � 3 � 2i � 6 � 13i

21.

� 3 � 3�2 i

��2 � ��8 � � �5 � ��50 � � �2 � 2�2 i � 5 � 5�2 i

22.

� 4

�8 ���18 � � �4 � 3�2 i� � 8 � 3�2 i � 4 � 3�2 i 23.

� �14 � 20i

13i � �14 � 7i� � 13i � 14 � 7i

24. 22 � ��5 � 8i� � 10i � 17 � 18i 25.

� 16 �

76i

� �96 �

156 i �

106 �

226 i

��32 �

52i� � �5

3 �113 i� � �

32 �

52i �

53 �

113 i

26. �1.6 � 3.2i� � ��5.8 � 4.3i� � �4.2 � 7.5i 27.

� 3 � i � 2 � 5 � i

�1 � i��3 � 2i� � 3 � 2i � 3i � 2i2

28.

� 12 � 22i � 6 � 6 � 22i

�6 � 2i��2 � 3i� � 12 � 18i � 4i � 6i2 29.

� 12 � 30i

6i�5 � 2i� � 30i � 12i2 � 30i � 12

30.

� 32 � 72i

�8i�9 � 4i� � �72i � 32i2 31.

� 14 � 10 � 24

��14 � �10 i���14 � �10 i� � 14 � 10i2

Page 57: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.5 Complex Numbers 123

32.

� 3 � 15 � 18

� 3 � 15��1�

��3 � �15 i���3 � �15 i� � 3 � 15i2 33.

� �9 � 40i

� 16 � 40i � 25

�4 � 5i�2 � 16 � 40i � 25i 2

34.

� �5 � 12i

� 4 � 9 � 12i

�2 � 3i�2 � 4 � 12i � 9i2 35.

� �10

� 4 � 12i � 9 � 4 � 12i � 9

�2 � 3i�2 � �2 � 3i�2 � 4 � 12i � 9i2 � 4 � 12i � 9i2

36.

� �8i

� 1 � 4i � 4i2 � 1 � 4i � 4i2

�1 � 2i�2 � �1 � 2i�2 � 1 � 4i � 4i2 � �1 � 4i � 4i2� 37. The complex conjugate of is

�6 � 3i��6 � 3i� � 36 � �3i�2 � 36 � 9 � 45

6 � 3i.6 � 3i

38. The complex conjugate of is

� 193

� 49 � ��144�

�7 � 12i��7 � 12i� � 49 � 144i2

7 � 12i.7 � 12i 39. The complex conjugate of is

� 1 � 5 � 6

��1 � �5i���1 � �5i� � ��1�2 � ��5i�2

�1 � �5i.�1 � �5i

40. The complex conjugate of is

� 11

� 9 � ��2�

��3 � �2 i���3 � �2 i� � 9 � 2i2

�3 � �2 i.�3 � �2 i 41. The complex conjugate of is

�2�5i���2�5i� � �20i2 � 20

�2�5i.��20 � 2�5i

42. The complex conjugate of is

��15 i����15 i� � �15i2 � ���15� � 15

��15 i.��15 � �15 i 43. The complex conjugate of is

��8���8� � 8

�8.�8

44. The complex conjugate of is

� 9 � 4�2

�1 � �8��1 � �8� � 1 � 2�8 � 8

1 � �8.1 � �8 45.5i

�5i

��i�i

��5i

1� �5i

46. �14

2i�

�2i

�2i�

28i

�4i 2�

28i

4� 7i 47.

�2�4 � 5i�16 � 25

�8 � 10i

41�

841

�1041

i

2

4 � 5i�

24 � 5i

�4 � 5i4 � 5i

48.5

1 � i�

1 � i

1 � i�

5 � 5i

1 � i2�

5 � 5i

2�

5

2�

5

2i 49.

�9 � 6i � i 2

9 � 1�

8 � 6i10

�45

�35

i

3 � i3 � i

�3 � i3 � i

�3 � i3 � i

50.

�20 � 5i

5�

20

5�

5

5i � 4 � i

6 � 7i

1 � 2i�

1 � 2i

1 � 2i�

6 � 12i � 7i � 14i2

1 � 4i251.

��6i � 5i 2

1� �5 � 6i

6 � 5i

i�

6 � 5ii

��i�i

Page 58: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

124 Chapter 1 Equations, Inequalities, and Mathematical Modeling

52.8 � 16i

2i�

�2i�2i

��16i � 32i2

�4i2� 8 � 4i

53.

� �1201681

�27

1681i

��27i � 120i2

81 � 1600�

�120 � 27i1681

3i

�4 � 5i�2 �3i

16 � 40i � 25i2�

3i�9 � 40i

��9 � 40i�9 � 40i

54.

�60 � 25i

169�

60169

�25

169i

��25i � 60i2

25 � 144i2

�5i

�5 � 12i�

�5 � 12i�5 � 12i

5i

�2 � 3i�2 �5i

4 � 12i � 9i2

55.

� �1

2�

5

2i

��1 � 5i

2

�2 � 2i � 3 � 3i

1 � 1

2

1 � i�

3

1 � i�

2�1 � i� � 3�1 � i��1 � i��1 � i�

56.

�12

5�

9

5i

�12 � 9i

5

�4i � 2i2 � 10 � 5i

4 � i2

2i

2 � i�

5

2 � i�

2i�2 � i��2 � i��2 � i�

�5�2 � i�

�2 � i��2 � i�

57.

�62 � 297i

949�

62

949�

297

949i

��100 � 297i � 162

949

��100 � 72i � 225i � 162i2

625 � 324

��4 � 9i

25 � 18i�

25 � 18i

25 � 18i

�4i2 � 9i

9 � 18i � 16

�3i � 8i2 � 6i � 4i2

9 � 24i � 6i � 16i2

i

3 � 2i�

2i

3 � 8i�

i�3 � 8i� � 2i�3 � 2i��3 � 2i��3 � 8i� 58.

�5

17�

20

17i

�5 � 20i

1 � 16i2

�5

1 � 4i�

1 � 4i

1 � 4i

�4 � i � 4i � i2 � 3i

4i � i2

1 � i

i�

3

4 � i�

�1 � i��4 � i� � 3i

i�4 � i�

59.

� �2�3

��6 � ��2 � ��6i���2i� � �12i 2 � �2�3 ���1� 60.

� �50 i2 � 5�2��1� � �5�2

��5 � ��10 � ��5 i���10 i�

61. ���10 �2� ��10 i �2

� 10i2 � �10 62. ���75 �2� ��75 i�2

� 75i2 � �75

63.

� �21 � 5�2 � � �7�5 � 3�10 �i � �21 � �50 � � �7�5 � 3�10 �i � 21 � 3�10 i � 7�5 i � �50 i2

�3 � ��5 ��7 � ��10 � � �3 � �5 i��7 � �10 i�

Page 59: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.5 Complex Numbers 125

64.

� �2 � 4�6i

� 4 � 6 � 4�6i

� 4 � 2�6i � 2�6i � 6��1�

� 4 � 2�6i � 2�6i � 6i2

�2 � ��6 �2� �2 � �6i��2 � �6i� 65.

� 1 ± i

�2 ± 2i

2

�2 ± ��4

2

x ����2� ± ���2�2 � 4�1��2�

2�1�

x2 � 2x � 2 � 0; a � 1, b � �2, c � 2

66.

� �3 ± i

��6 ± ��4

2

x ��6 ± �62 � 4�1��10�

2�1�

a � 1, b � 6, c � 10x2 � 6x � 10 � 0; 67.

��16 ± 4i

8� �2 ±

1

2i

��16 ± ��16

8

x ��16 ± ��16�2 � 4�4��17�

2�4�

4x2 � 16x � 17 � 0; a � 4, b � 16, c � 17

68.

�1

36i

18�

1

3± 2i

�6 ± ��1296

18

x ����6� ± ���6�2 � 4�9��37�

2�9�

9x2 � 6x � 37 � 0; a � 9, b � �6, c � 37 69.

x � �12

8� �

3

2 or x � �

20

8� �

5

2

��16 ± �16

8�

�16 ± 4

8

x ��16 ± ��16�2 � 4�4��15�

2�4�

4x2 � 16x � 15 � 0; a � 4, b � 16, c � 15

70.

�1

8±�11

8i

�4 ± 4�11 i

32

�4 ± ��176

32

t ����4� ± ���4�2 � 4�16��3�

2�16�

16t2 � 4t � 3 � 0; a � 16, b � �4, c � 3 71. Multiply both sides by 2.

�12 ± 6�2i

6� 2 ± �2i

�12 ± ��72

6

x ����12� ± ���12�2 � 4�3��18�

2�3�

3x2 � 12x � 18 � 0

32

x2 � 6x � 9 � 0

72.

�37

±�3414

i

�12 ± 2i�34

28

�12 ± ��136

28

x ����12� ± ���12�2 � 4�14��5�

2�14�

14x2 � 12x � 5 � 0; a � 14, b � �12, c � 5

78

x2 �34

x �5

16� 0 73. Multiply both sides by 5.

�5 ± 5�15

7�

57

±5�15

7

�10 ± �1500

14�

10 ± 10�1514

x ����10� ± ���10�2 � 4�7���50�

2�7�

7x2 � 10x � 50 � 0

1.4x2 � 2x � 10 � 0

Page 60: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

126 Chapter 1 Equations, Inequalities, and Mathematical Modeling

74.

�3 ± ��207

9�

3 ± 3i�239

�13

±�23

3i

x ����3� ± ���3�2 � 4�4.5��12�

2�4.5�

4.5x2 � 3x � 12 � 0; a � 4.5, b � �3, c � 12 75.

� �1 � 6i

� 6i � 1

� �6��1�i � ��1�

�6i 3 � i 2 � �6i2i � i2

76. 4i2 � 2i3 � �4 � 2i 77.

� �5i

� �5��1���1�i

�5i5 � �5i2i2i 78. ��i�3 � ��1��i3� � ��1���i� � i

79.

� �375�3 i

� 125�3�3 � ��1�i

� 53��3 �3i 3

���75 �3 � �5�3 i�3 80. ���2 �6� ��2 i�6

� 8i 6 � 8i4i2 � �8

81.1

i3�

1

�i�

1

�i�

i

i�

i

�i2�

i

1� i 82.

1

�2i�3�

1

8i3�

1

�8i�

8i

8i�

8i

�64i2�

1

8i

84. (a)

(b)

(c)

� 8

� �1 � 3�3 i � 9 � 3�3 i

� �1 � 3�3 i � 9i2 � 3�3 i3

��1 � �3 i�3 � ��1�3 � 3��1�2 ���3 i� � 3��1����3 i� 2 � � � �3 i�3

� 8

� �1 � 3�3 i � 9 � 3�3i

� �1 � 3�3 i � 9i2 � 3�3 i3

��1 � �3i�3 � ��1�3 � 3��1�2 ��3i� � 3��1���3i�2 � ��3i� 3

�2�3 � 8

85. (a)

(b)

(c)

(d) ��2i�4 � ��2�4i 4 � 16�1� � 16

�2i�4 � 24i 4 � 16�1� � 16

��2�4 � 16

24 � 16 86. (a)

(b)

(c)

(d) i67 � �i 4�16�i 3� � �1���i� � �i

i50 � �i 4�12�i 2� � �1���1� � �1

� i

� �1�6i

i 25 � �i 4�6 � i

� 1

� �1�10

i 40 � �i 4�10

83. (a)

(b)

�11,240 � 4630i

877�

11,240877

�4630877

i z � �340 � 230i29 � 6i ��29 � 6i

29 � 6i�

�29 � 6i

340 � 230i�

20 � 10i � 9 � 16i�9 � 16i��20 � 10i��

19 � 16i

�1

20 � 10i 1z

�1z1

�1z2

z1 � 9 � 16i, z2 � 20 � 10i

Page 61: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.5 Complex Numbers 127

87. False, if then

That is, if the complex number is real, the number equalsits conjugate.

a � bi � a � bi � a.b � 0 88. True

56 � 56

36 � 6 � 14 �?

56

��i�6 �4� ��i�6 �2

� 14 �?

56

x 4 � x2 � 14 � 56

89. False

� 1 � ��1� � 1 � i � i � 1

� �1�11 � �1�37��1� � �1�18��1� � �1�27�i� � �1�15�i�

i44 � i150 � i 74 � i109 � i61 � �i4�11 � �i4�37�i2� � �i4�18�i2� � �i4�27�i� � �i4�15�i�

90. ��6��6 � �6i�6i � 6i2 � �6

91.

The complex conjugate of this product is

The product of the complex conjugates is:

Thus, the complex conjugate of the product of two complex numbers is the product of their complex conjugates.

� �a1a2 � b1b2� � �a1b2 � a2b1�i

�a1 � b1i��a2 � b2i� � a1a2 � a1b2i � a2b1i � b1b2i2

�a1a2 � b1b2� � �a1b2 � a2b1�i.

� �a1a2 � b1b2� � �a1b2 � a2b1�i

�a1 � b1i��a2 � b2i� � a1a2 � a1b2i � a2b1i � b1b2i2

92.

The complex conjugate of this sum is

The sum of the complex conjugates is

Thus, the complex conjugate of the sum of two complex numbers is the sum of their complex conjugates.

�a1 � b1i� � �a2 � b2i� � �a1 � a2� � �b1 � b2�i.

�a1 � a2� � �b1 � b2�i.

�a1 � b1i� � �a2 � b2i� � �a1 � a2� � �b1 � b2�i

93. �4 � 3x� � �8 � 6x � x2� � �x2 � 3x � 12

94.

� x3 � x2 � 2x � 6

�x3 � 3x2� � �6 � 2x � 4x2� � x3 � 3x2 � 6 � 2x � 4x2 95.

� 3x2 �232 x � 2

�3x �12��x � 4� � 3x2 � 12x �

12x � 2

96.

� 4x2 � 20x � 25

�2x � 5�2 � �2x�2 � 2�2x��5� � �5�2 97.

x � �31

�x � 31

�x � 12 � 19 98.

x � 14

�3x � �42

8 � 3x � �34

99.

x �272

2x � 27

2x � 27 � 0

20x � 24 � 18x � 3 � 0

4�5x � 6� � 3�6x � 1� � 0 100.

x �40

�30� �

43

�30x � 40

5x � 15x � 55 � 20x � 15

5�x � �3x � 11�� � 20x � 15 101.

a �1

2�3V

�b�

�3V�b

2�b

� 3V

4�b� a

3V

4�b� a2

3V � 4�a2b

V �4

3�a2b

Page 62: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

128 Chapter 1 Equations, Inequalities, and Mathematical Modeling

102.

r ���m1m2

F�

��m1m2

�F��F�F

���m1m2F

F

r2 � �m1m2

F

F � �m1m2

r2103. Let liters withdrawn and replaced.

x � 1 liter

0.50x � 0.50

2.50 � 0.50x � 1.00x � 3.00

0.50�5 � x� � 1.00x � 0.60�5�

x � #

Section 1.6 Other Types of Equations

■ You should be able to solve certain types of nonlinear or nonquadratic equations by rewriting them in a form in whichyou can factor, extract square roots, complete the square, or use the Quadratic Formula.

■ For equations involving radicals or rational exponents, raise both sides to the same power.

■ For equations that are of the quadratic type, use either factoring, the Quadratic Formula,or completing the square.

■ For equations with fractions, multiply both sides by the least common denominator to clear the fractions.

■ For equations involving absolute value, remember that the expression inside the absolute value can be positive or negative.

■ Always check for extraneous solutions.

au2 � bu � c � 0, a � 0,

1.

2x2 � 9 � 0 ⇒ x � ±3�2

2

2x2 � 0 ⇒ x � 0

2x2�2x2 � 9� � 0

4x4 � 18x2 � 0

2.

2x � 5 � 0 ⇒ x �52

2x � 5 � 0 ⇒ x � �52

5x � 0 ⇒ x � 0

5x�2x � 5��2x � 5� � 0

5x�4x2 � 25� � 0

20x3 � 125x � 0 3.

2x � 3 � 0 ⇒ x � 3

2x � 3 � 0 ⇒ x � �3

x2 � 9 � 0 ⇒ x � ±3i

�x2 � 9��x � 3��x � 3� � 0

x4 � 81 � 0

4.

x2 � 2x � 4 � 0 ⇒ x � 1 ± �3 i

x � 2 � 0 ⇒ x � �2

x2 � 2x � 4 � 0 ⇒ x � �1 ± �3 i

x � 2 � 0 ⇒ x � 2

�x � 2��x2 � 2x � 4��x � 2��x2 � 2x � 4� � 0

�x3 � 8��x3 � 8� � 0

x6 � 64 � 0

Vocabulary Check

1. polynomial 2. extraneous 3. quadratic type

Page 63: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.6 Other Types of Equations 129

5.

x2 � 6x � 36 � 0 ⇒ x � 3 ± 3�3i

x � 6 � 0 ⇒ x � �6

�x � 6��x2 � 6x � 36� � 0

x3 � 63 � 0

x3 � 216 � 0

6.

� �43

±4�3 i

3

� �43

±24�3 i

18

��2418

±��1728

18

��24 ± ��1728

18

9x2 � 24x � 64 � 0 ⇒ x ��24 ± ��24�2 � 4�9��64�

2�9�

3x � 8 � 0 ⇒ x �8

3

�3x � 8��9x2 � 24x � 64� � 0

27x3 � 512 � 0

7.

x � 3 � 0 ⇒ x � �3

5x � 0 ⇒ x � 0

5x�x � 3�2 � 0

5x�x2 � 6x � 9� � 0

5x3 � 30x2 � 45x � 0 8.

3x � 4 � 0 ⇒ x �43

x2 � 0 ⇒ x � 0

x2�3x � 4�2 � 0

x2�9x2 � 24x � 16� � 0

9x4 � 24x3 � 16x2 � 0

9.

x � 1 � 0 ⇒ x � 1

x � 1 � 0 ⇒ x � �1

x � 3 � 0 ⇒ x � 3

�x � 3��x � 1��x � 1� � 0

�x � 3��x2 � 1� � 0

x2�x � 3� � �x � 3� � 0

x3 � 3x2 � x � 3 � 0 10.

x2 � 3 � 0 ⇒ x � ±�3 i

x � 2 � 0 ⇒ x � �2

�x � 2��x2 � 3� � 0

x2�x � 2� � 3�x � 2� � 0

x3 � 2x2 � 3x � 6 � 0

11.

x2 � x � 1 � 0 ⇒ x �1

2±�3

2i

x � 1 � 0 ⇒ x � �1

x � 1 � 0 ⇒ x � 1

�x � 1��x � 1��x2 � x � 1� � 0

�x � 1��x3 � 1� � 0

x3�x � 1� � �x � 1� � 0

x4 � x3 � x � 1 � 0

Page 64: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

130 Chapter 1 Equations, Inequalities, and Mathematical Modeling

12.

x � 2 � 0 ⇒ x � �2

x2 � 2x � 4 � 0 ⇒ x � �1 ± �3 i

x � 2 � 0 ⇒ x � 2

�x � 2��x2 � 2x � 4��x � 2� � 0

�x3 � 8��x � 2� � 0

x3�x � 2� � 8�x � 2� � 0

x4 � 2x3 � 8x � 16 � 0 13.

x � 1 � 0 ⇒ x � 1

x � 1 � 0 ⇒ x � �1

x � �3 � 0 ⇒ x � �3

x � �3 � 0 ⇒ x � ��3

�x � �3 ��x � �3 ��x � 1��x � 1� � 0

�x2 � 3��x2 � 1� � 0

x4 � 4x2 � 3 � 0

14.

x � 2 � 0 ⇒ x � 2

x � 2 � 0 ⇒ x � �2

x2 � 9 � 0 ⇒ x � ±3i

�x2 � 9��x � 2��x � 2� � 0

�x2 � 9��x2 � 4� � 0

x4 � 5x2 � 36 � 0 15.

1x � 4 � 0 ⇒ x � 4

1x � 4 � 0 ⇒ x � �4

2x � 1 � 0 ⇒ x �12

2x � 1 � 0 ⇒ x � �12

�2x � 1��2x � 1��x � 4��x � 4� � 0

�4x2 � 1��x2 � 16� � 0

4x4 � 65x2 � 16 � 0

16.

t2 � 1 � 0 ⇒ t � ± i

6t � �7 � 0 ⇒ t ��7

6

6t � �7 � 0 ⇒ t � ��7

6

�6t � �7 ��6t � �7 ��t2 � 1� � 0

�36t2 � 7��t2 � 1� � 0

36t4 � 29t2 � 7 � 0

17.

x2 � x � 1 � 0 ⇒ x � �1

2 ±

�3

2i

x � 1 � 0 ⇒ x � 1

x2 � 2x � 4 � 0 ⇒ x � 1 ± �3i

x � 2 � 0 ⇒ x � �2

�x � 2��x2 � 2x � 4��x � 1��x2 � x � 1� � 0

�x3 � 8��x3 � 1� � 0

x6 � 7x3 � 8 � 0

18.

x2 � x � 1 � 0 ⇒ x �1 ± �3 i

2

x � 1 � 0 ⇒ x � �1

x2 � 3�2x � � 3�2�2� 0 ⇒ x �

1 ± �3 i

22�3

x � 3�2 � 0 ⇒ x � � 3�2

�x � 3�2 ��x2 � 3�2x � �3�2 �2��x � 1��x2 � x � 1� � 0

�x3 � 2��x3 � 1� � 0

x6 � 3x3 � 2 � 0

Page 65: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.6 Other Types of Equations 131

19.

1 � 5x � 0 ⇒ x � �1

5

1 � 3x � 0 ⇒ x � �1

3

�1 � 3x��1 � 5x� � 0

1 � 8x � 15x2 � 0

1

x2�

8

x� 15 � 0 20.

Let

x

x � 1� �

3

2 ⇒ x � �

3

5

x

x � 1�

2

3 ⇒ x � 2

2u � 3 � 0 ⇒ u � �3

2

3u � 2 � 0 ⇒ u �2

3

�3u � 2��2u � 3� � 0

6u2 � 5u � 6 � 0

u � x��x � 1�.

6� x

x � 1�2

� 5� x

x � 1� � 6 � 0

21.

��x � �5 is not a solution.� �x �

12 ⇒ x �

14

�2�x � 1���x � 5� � 0

2��x �2� 9�x � 5 � 0

2x � 9�x � 5 � 0

2x � 9�x � 5 22.

Let

is not a solution.

�x �32 ⇒ x �

94

�x � �13

2u � 3 � 0 ⇒ u �32

3u � 1 � 0 ⇒ u � �13

�3u � 1��2u � 3� � 0

6u2 � 7u � 3 � 0

u � �x.

6x � 7�x � 3 � 0

23.

x1�3 � 1 � 0 ⇒ x1�3 � 1 ⇒ x � �1�3 � 1

2x1�3 � 5 � 0 ⇒ x1�3 � �52 ⇒ x � �� 5

2�3� �

1258

�2x1�3 � 5��x1�3 � 1� � 0

2�x1�3�2 � 3x1�3 � 5 � 0

2x2�3 � 3x1�3 � 5 � 0

3x1�3 � 2x2�3 � 5

24.

t � �6427

3t1�3 � 4 � 0 ⇒ t1�3 � �43

�3t1�3 � 4�2 � 0

9t2�3 � 24t1�3 � 16 � 0

25.

(a)

(b) x-intercepts: ��1, 0�, �0, 0�, �3, 0�

−9 9

−7

5

y � x3 � 2x2 � 3x

(c)

(d) The intercepts of the graph are the same as the solutionsto the equation.

x-

x � 3 � 0 ⇒ x � 3

x � 1 � 0 ⇒ x � �1

x � 0

0 � x�x � 1��x � 3�

0 � x3 � 2x2 � 3x

Page 66: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

132 Chapter 1 Equations, Inequalities, and Mathematical Modeling

26. (a)

(b) x-intercepts:

(c)

(d) The x-intercepts and the solutions are the same.

x � 0, 32, 6

0 � x � 6 ⇒ x � 6

0 � 2x � 3 ⇒ x �32

0 � x2 ⇒ x � 0

� x2(2x � 3)(x � 6)

� x2(2x2 � 15x � 18)

0 � 2x4 � 15x3 � 18x2

y � 2x4 � 15x3 � 18x2

�0, 0�, �32, 0�, �6, 0�

−3 9

−200

40 27.

(a)

(b) x-intercepts:

(c)

(d) The x-intercepts of the graph are the same as thesolutions to the equation.

x � 3 � 0 ⇒ x � 3

x � 3 � 0 ⇒ x � �3

x � 1 � 0 ⇒ x � 1

x � 1 � 0 ⇒ x � �1

0 � �x � 1��x � 1��x � 3��x � 3�

0 � �x2 � 1��x2 � 9�

0 � x4 � 10x2 � 9

�±1, 0�, �±3, 0�

−5 5

20

−20

y � x4 � 10x2 � 9

28. (a)

(b) x-intercepts:

(d) The x-intercepts and the solutions are the same.

��5, 0�, �5, 0���2, 0�, �2, 0�,

−10 10

150

−150

(c)

x � ±5, ±2

0 � x � 5 ⇒ x � 5

0 � x � 5 ⇒ x � �5

0 � x � 2 ⇒ x � 2

0 � x � 2 ⇒ x � �2

� �x � 2��x � 2��x � 5��x � 5�

� �x2 � 4��x2 � 25�

0 � x4 � 29x2 � 100

y � x4 � 29x2 � 100

29.

x � 50

2x � 100

�2x � 10

�2x � 10 � 0 30.

x �916

16x � 9

4�x � 3

4�x � 3 � 0 31.

x � 26

x � 10 � 16

�x � 10 � 4

�x � 10 � 4 � 0

32.

x � �4

5 � x � 9

�5 � x � 3

�5 � x � 3 � 0 33.

x � �16

2x � �32

2x � 5 � �27

33�2x � 5 � �3

3�2x � 5 � 3 � 0 34.

x �1243

3x � 124

3x � 1 � 125

3�3x � 1 � 5

3�3x � 1 � 5 � 0

Page 67: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.6 Other Types of Equations 133

35.

x � 2 � 0 ⇒ x � 2

x � 5 � 0 ⇒ x � �5

�x � 5��x � 2� � 0

x2 � 3x � 10 � 0

16 � 8x � x2 � 26 � 11x

4 � x � �26 � 11x

��26 � 11x � 4 � x 36.

0 � x � 2 ⇒ x � �2

0 � x � 3 ⇒ x � 3

0 � �x � 3��x � 2�

0 � x2 � x � 6

31 � 9x � 25 � 10x � x2

�31 � 9x � 5 � x

x � �31 � 9x � 5

37.

x � 0

�2x � 0

x � 1 � 3x � 1

�x � 1 � �3x � 1 38.

No solution

5 � �5

x � 5 � x � 5

�x � 5 � �x � 5

39.

9 � x

4 � x � 5

2 � �x � 5

4 � 2�x � 5

x � 1 � 2�x � 5 � x � 5

��x �2� �1 � �x � 5 � 2

�x � 1 � �x � 5

�x � �x � 5 � 1 40.

36 � x

16 � x � 20

4 � �x � 20

�80 � �20�x � 20

x � 100 � 20�x � 20 � x � 20

��x �2� �10 � �x � 20�2

�x � 10 � �x � 20

�x � �x � 20 � 10

41.

1014 � x

101 � 4x

81 � 4x � 20

81 � 4�x � 5�

9 � 2�x � 5

�90 � �20�x � 5

x � 5 � 100 � 20�x � 5 � x � 5

��x � 5 �2� �10 � �x � 5 �2

�x � 5 � 10 � �x � 5

�x � 5 � �x � 5 � 10 42.

x � 1 � 0 ⇒ x � �1, extraneous

x � 3 � 0 ⇒ x � 3

�x � 3��x � 1� � 0

x2 � 2x � 3 � 0

x2 � 2x � 3

x � �2x � 3

2x � 2�2x � 3

4�x � 1� � 1 � 2�2x � 3 � 2x � 3

�2�x � 1 �2� �1 � �2x � 3 �2

2�x � 1 � 1 � �2x � 3

2�x � 1 � �2x � 3 � 1

Page 68: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

134 Chapter 1 Equations, Inequalities, and Mathematical Modeling

43.

x � 14 � 0 ⇒ x � 14

x � 2 � 0 ⇒ x � 2, extraneous

�x � 2��x � 14� � 0

x2 � 16x � 28 � 0

x2 � 8x � 16 � 8x � 12

x2 � 8x � 16 � 4�2x � 3�

�x � 4�2 � �2�2x � 3�2

x � 4 � 2�2x � 3

�x � 4 � �2�2x � 3

x � 2 � 2x � 2 � 2�2x � 3

x � 2 � 2x � 3 � 2�2x � 3 � 1

��x � 2�2� ��2x � 3 � 1�2

�x � 2 � �2x � 3 � 1

�x � 2 � �2x � 3 � �1 44.

x � 7 � 0 ⇒ x � 7

25x � 79 � 0 ⇒ x �7925, extraneous

�25x � 79��x � 7� � 0

25x2 � 254x � 553 � 0

100x2 � 1016x � 2212 � 0

100x2 � 800x � 1600 � 36�6x � 17�

�10x � 40�2 � �6�6x � 17 �2

10x � 40 � 6�6x � 17

16�x � 3� � 9 � 6�6x � 17 � 6x � 17

�4�x � 3 �2� �3 � �6x � 17 �2

4�x � 3 � 3 � �6x � 17

4�x � 3 � �6x � 17 � 3

45.

x � 5 � 4 � 9

x � 5 � 3�64

�x � 5�3 � 82

�x � 5�3�2 � 8 46.

x � �3 � 4 � 1

x � 3 � 3�64

�x � 3�3 � 64

�x � 3�3 � 82

�x � 3�3�2 � 8 47.

x � �3 ± 16�2

x � 3 � ±�512

x � 3 � ±�83

�x � 3�2 � 83

�x � 3�2�3 � 8

48.

x � �2 ± 27 � �29, 25

x � 2 � ±�729

�x � 2�2 � 93

�x � 2�2�3 � 9 49.

x � ±�14

x2 � 14

x2 � 5 � 9

x2 � 5 � 3�272

�x2 � 5�3 � 272

�x2 � 5�3�2 � 27

50.

x ����1� ± ���1�2 � 4�1���31�

2�1� �1 ± �125

2�

1 ± 5�52

x2 � x � 31 � 0

x2 � x � 22 � 9

x2 � x � 22 � 272�3

�x2 � x � 22�3�2 � 27

51.

5x � 2 � 0 ⇒ x �25, extraneous

�x � 1�1�2 � 0 ⇒ x � 1 � 0 ⇒ x � 1

�x � 1�1�2�5x � 2� � 0

�x � 1�1�2�3x � 2�x � 1�� � 0

3x�x � 1�1�2 � 2�x � 1�3�2 � 0 52.

5x � 3 � 0 ⇒ x �35

x � 1 � 0 ⇒ x � 1

2x � 0 ⇒ x � 0

2x�x � 1�1�3�5x � 3� � 0

2x�x � 1�1�3�2x � 3�x � 1�� � 0

2x�2x�x � 1�1�3 � 3�x � 1�4�3� � 0

4x2�x � 1�1�3 � 6x�x � 1�4�3 � 0

Page 69: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.6 Other Types of Equations 135

53.

(a)

(b) x-intercepts:

(c)

(d) The x-intercepts of the graph are the same as the solutions to the equation.

x � 6 � 0 ⇒ x � 6

x � 5 � 0 ⇒ x � 5

�x � 5��x � 6� � 0

x2 � 11x � 30 � 0

x2 � 11x � 30

x � �11x � 30

0 � �11x � 30 � x

�5, 0�, �6, 0�

−1 11

−6

2

y � �11x � 30 � x 54. (a)

(b) x-intercept:

(c)

(d) The x-intercept and the solution are the same.

x �32

0 � 2x � 3 ⇒ x � 32

0 � 2x � 5 ⇒ x � �52, extraneous

0 � �2x � 5��2x � 3�

0 � 4x2 � 4x � 15

15 � 4x � 4x2

�15 � 4x � 2x

0 � 2x � �15 � 4x

y � 2x � �15 � 4x

�32, 0�

−4 8

−5

3

55.

(a)

(b) x-intercepts:

(c)

(d) The x-intercepts of the graph are the same as thesolutions to the equation.

x � 4 � 0 ⇒ x � 4

x � 0

x�x � 4� � 0

x2 � 4x � 0

x2 � 16x � 64 � 20x � 64

x2 � 16x � 64 � 4�5x � 16�

�x � 8�2 � �2�5x � 16�2

x � 8 � 2�5x � 16

2x � 16 � 4�5x � 16

7x � 36 � 5x � 20 � 4�5x � 16

7x � 36 � 4 � 4�5x � 16 � 5x � 16

��7x � 36 �2� �2 � �5x � 16 �2

�7x � 36 � 2 � �5x � 16

��7x � 36 � ��5x � 16 � 2

0 � �7x � 36 � �5x � 16 � 2

�0, 0�, �4, 0�−0.5

0.5

5−3

y � �7x � 36 � �5x � 16 � 2 56. (a)

(b) x-intercept:

(c)

(d) The x-intercept and the solution are the same.

x � 4

0 � �x � 2 ⇒ x � 4

0 � 3�x � 2 ⇒ x �4

9, extraneous

0 � �3�x � 2���x � 2�0 � 3x � 4 � 4�x

0 � �x�3�x �4

�x� 4�

0 � 3�x �4

�x� 4

y � 3�x �4

�x� 4

(4, 0)

−2 10

−4

4

Page 70: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

136 Chapter 1 Equations, Inequalities, and Mathematical Modeling

57.

x � 2 � 0 ⇒ x � 2

2x � 3 � 0 ⇒ x � �3

2

�2x � 3��x � 2� � 0

2x2 � x � 6 � 0

2x2 � 6 � x

�2x��x� � �2x��3

x� � �2x��1

2�

x �3

x�

1

258.

x � 2 � 0 ⇒ x � 2

x � 12 � 0 ⇒ x � �12

�x � 12��x � 2� � 0

x2 � 10x � 24 � 0

24 � 10x � x2

�6x�4

x� �6x�

5

3� �6x�

x

6

4

x�

5

3�

x

6

59.

x ��3 ± ��3�2 � 4�3���1�

2�3��

�3 ± �21

6

a � 3, b � 3, c � �1

0 � 3x2 � 3x � 1

1 � 3x2 � 3x

x � 1 � x � 3x�x � 1�

x�x � 1�1

x� x�x � 1�

1

x � 1� x�x � 1��3�

1

x�

1

x � 1� 3 60.

x � 3 � 0 ⇒ x � �3

x � 1 � 0 ⇒ x � 1

�x � 1��x � 3� � 0

x2 � 2x � 3 � 0

4x � 8 � 3x � 3 � x2 � 3x � 2

4�x � 2� � 3�x � 1� � �x � 1��x � 2�

4

x � 1�

3

x � 2� 1

61.

x � 4 � 0 ⇒ x � 4

x � 5 � 0 ⇒ x � �5

0 � �x � 5��x � 4�

0 � x2 � x � 20

20 � x � x2

20 � x

x� x 62.

x � 1 � 0 ⇒ x � �1

4x � 3 � 0 ⇒ x �3

4

�4x � 3��x � 1� � 0

4x2 � x � 3 � 0

4x2 � x � 3

�x�4x � �x�1 � �x�3

x

4x � 1 �3

x

63.

�2 ± �124

6�

2 ± 2�31

6�

1 ± �31

3

x ����2� ± ���2�2 � 4�3���10�

2�3�

a � 3, b � �2, c � �10

3x2 � 2x � 10 � 0

x � x � 2 � 3x2 � 12

�x � 2��x � 2�x

x2 � 4� �x � 2��x � 2�

1

x � 2� 3�x � 2��x � 2�

x

x2 � 4�

1

x � 2� 3 64.

x � 1 � 0 ⇒ x � 1

x � 1 � 0 ⇒ x � �1

�x � 1��x � 1� � 0

x2 � 1 � 0

x2 � 3x � 2 � 3x � 3 � 0

�x � 2��x � 1) � 3�x � 1� � 0

3�x � 2�x � 1

3� 3�x � 2�

x � 1

x � 2� 0

x � 1

3�

x � 1

x � 2� 0

Page 71: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.6 Other Types of Equations 137

65.

��2x � 1� � 5 ⇒ x � �2

2x � 1 � 5 ⇒ x � 3

2x � 1 � 5 66.

�3x � 2 � 7 ⇒ x � �3

��3x � 2� � 7

3x � 2 � 7 ⇒ x �53

3x � 2 � 7

67.

First equation:

Only are solutions to the original equation. and are extraneous.x � 1x � ��3 x � �3 and x � �3

x � ±�3

x2 � 3 � 0

x � x2 � x � 3

x � x2 � x � 3

Second equation:

x � 3 � 0 ⇒ x � �3

x � 1 � 0 ⇒ x � 1

�x � 1��x � 3� � 0

x2 � 2x � 3 � 0

�x � x2 � x � 3

68.

First equation:

The solutions to the original equation are x � ±3 and x � �6.

x � 6 � 0 ⇒ x � �6

x � 3 � 0 ⇒ x � 3

�x � 3��x � 6� � 0

x2 � 3x � 18 � 0

x2 � 6x � 3x � 18

x2 � 6x � 3x � 18

Second equation:

0 � x � 6 ⇒ x � �6

0 � x � 3 ⇒ x � �3

0 � �x � 3��x � 6�

0 � x2 � 9x � 18

��x2 � 6x� � 3x � 18

69.

First equation:

Only are solutions to the original equation. and are extraneous.x ��1 � �17

2x � �2x � 3 and x �

�1 � �17

2

x � 2 � 0 ⇒ x � �2

x � 3 � 0 ⇒ x � 3

�x � 3��x � 2� � 0

x2 � x � 6 � 0

x � 1 � x2 � 5

x � 1 � x2 � 5

Second equation:

x ��1 ± �17

2

x2 � x � 4 � 0

�x � 1 � x2 � 5

��x � 1� � x2 � 5

70.

First equation:

The solutions to the original equation are is extraneous.x � 10 and x � �1. x � 1

0 � x � 10 ⇒ x � 10

0 � x � 1 ⇒ x � 1

0 � �x � 1��x � 10�

0 � x2 � 11x � 10

x � 10 � x2 � 10x

x � 10 � x2 � 10x

Second equation:

0 � x � 1 ⇒ x � �1

0 � x � 10 ⇒ x � 10

0 � �x � 10��x � 1�

0 � x2 � 9x � 10

��x � 10� � x2 � 10x

Page 72: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

138 Chapter 1 Equations, Inequalities, and Mathematical Modeling

71.

(a)

(b) x-intercept:

(c)

(d) The x-intercept of the graph is the same as the solution to the equation.

x � 1 � 0 ⇒ x � �1

0 � �x � 1�2

0 � x2 � 2x � 1

0 � �x2 � 2x � 1

0 � x � 1 � 4x � x2 � x

0 � �x � 1� � 4x � x�x � 1�

0 �1

x�

4

x � 1� 1

��1, 0�

4−4

−24

24

y �1

x�

4

x � 1� 1 72.

(a)

(b) x-intercept:

(c)

(d) The x-intercept and the solution are the same.

x � 2

0 � x � 2 ⇒ x � 2

0 � �x � 2��x � 2�

0 � x2 � 4x � 4

0 � x2 � x � 9 � 5x � 5

0 � x�x � 1� � �x � 1�9

x � 1� 5�x � 1�

0 � x �9

x � 1� 5

0 � x �9

x � 1� 5

�2, 0�

−15 15

−30

20

y � x �9

x � 1� 5

73.

(a)

(b) x-intercepts:

(c)

OR

OR

(d) The x-intercepts of the graph are the same asthe solutions to the equation.

x � �3

�x � 3

�x � 1 � 2 x � 1

��x � 1� � 2 x � 1 � 2

2 � x � 1 0 � x � 1 � 2

�1, 0�, ��3, 0�

−10 8

−4

8

y � x � 1 � 2 74.

(a)

(b) x-intercepts:

(c)

OR

(d) The x-intercepts and the solutions are the same.

x � 2 � 3 ⇒ x � 5

3 � x � 20 � x � 2 � 3

�5, 0�, ��1, 0�

−7 11

−5

7

y � x � 2 � 3

75.

Using the positive value for we have x � ±�1.5 � �29.13

6.4 ±1.038.x2,

x2 �1.5 ± �1.52 � 4�3.2���2.1�

2�3.2�

3.2x4 � 1.5x2 � 2.1 � 0

x � 5, �1

⇒ x � �1 �x � 2 � 3

��x � 2� � 3

Page 73: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.6 Other Types of Equations 139

76.

x � 3��4.15 � �289.0945

14.16 �1.143

x � 3��4.15 � �289.0945

14.16 0.968

��4.15 ± �289.0945

14.16

x3 ��4.15 ± ��4.15�2 � 4�7.08���9.6�

2�7.08�

a � 7.08, b � 4.15, c � �9.6

7.08x6 � 4.15x3 � 9.6 � 0 77.

Use the Quadratic Formula with and

Considering only the positive value for we have:

x 16.756.

�x 4.0934

�x ,

�x �6 ± �36 � 4�1.8���5.6�

2�1.8�

6 ± 8.7361

3.6

c � �5.6.b � �6,a � 1.8,

1.8��x �2� 6�x � 5.6 � 0

1.8x � 6�x � 5.6 � 0 Given equation

78.

x � ��8 � �6.4

8 �3

�2.280

x � ��8 � �6.4

8 �3

�0.320

x1�3 ��8 ± �82 � 4�4��3.6�

2�4�

a � 4, b � 8, c � 3.6

4x2�3 � 8x1�3 � 3.6 � 0 79. and 5

One possible equation is:

Any non-zero multiple of this equation would also havethese solutions.

x2 � 3x � 10 � 0

�x � 2��x � 5� � 0

�x � ��2���x � 5� � 0

�2

80.

Any non-zero multiple of this equation would also havethese solutions.

x3 � 8x2 � 15x � 0

x�x2 � 8x � 15� � 0

x�x � 3��x � 5� � 0

�x � 0��x � 3��x � 5� � 0

0, 3, 5 81. and

One possible equation is:

Any non-zero multiple of this equation would also havethese solutions.

21x2 � 31x � 42 � 0

�3x � 7��7x � 6� � 0

x �67 ⇒ 7x � 6 ⇒ 7x � 6 is a factor.

x � �73 ⇒ 3x � �7 ⇒ 3x � 7 is a factor.

67�

73

82.

Any non-zero multiple of this equation would also havethese solutions.

40x2 � 37x � 4 � 0

40x2 � 32x � 5x � 4 � 0

x2 �45x �

18x �

440 � 0

�x �18��x �

45� � 0

�x � ��18���x � ��4

5�� � 0

�18, �4

5 83. and 4

One possible equation is:

Any non-zero multiple of this equation would also havethese solutions.

x3 � 4x2 � 3x � 12 � 0

�x2 � 3��x � 4� � 0

�x � �3��x � �3��x � 4� � 0

�x � �3��x � ���3���x � 4� � 0

�3, ��3,

84.

Any non-zero multiple of this equation would also havethese solutions.

x2 � x�7 � 14 � 0

x2 � x�7 � 2x�7 � 2�7� � 0

�x � 2�7 ��x � �7 � � 0

2�7, ��7 85. and

One possible equation is:

Any non-zero multiple of this equation would also havethese solutions.

x4 � 1 � 0

�x2 � 1��x2 � 1� � 0

�x � 1��x � 1��x � i��x � i� � 0

�x � ��1���x � 1��x � i��x � ��i�� � 0

�i�1, 1, i,

Page 74: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

140 Chapter 1 Equations, Inequalities, and Mathematical Modeling

86.

x 4 � 20x2 � 576 � 0

�x2 � 16��x2 � 36� � 0

�x � 4i��x � 4i��x � 6��x � 6� � 0

4i, �4i, 6, �6

87. Let the number of students in the original group. Then, the original cost per student.

When six more students join the group, the cost per student becomes

Model:

Multiply both sides by to clear fractions.

Using the positive value for x we conclude that the original number was students.x � 34

x �90 ± ���90�2 � 4��15��20,400�

2��15��

90 ± 1110

�30

�15x2 � 90x � 20,400 � 0

2x �3400 � 15x��x � 6� � 3400x

�1700

x� 7.5��x � 6� � 1700

�Cost per student� � �Number of students� � Total cost

1700x

� 7.50.

1700x

�x �

88. Model:

Labels: Monthly rent

Number of students

Original cost per student

Cost per student

Equation:

The monthly rent is $900.

x � 900

x3

� 300

4x3

� x � 300

4x3

� 300 � x

�x3

� 75��4� � x

�x3

� 75

�x3

� 4

� x

�Cost perstudent � � �Number of

students � � �Monthlyrent � 89. Model:

Labels: Let average speed of the plane. Then wehave a travel time of If the averagespeed is increased by 40 mph, then

Now, we equate these two equations and solve for x.

Equation:

Using the positive value for x found by the Quadratic Formula, we have mph and

mph. The airspeed required to obtain the decrease in travel time is 191.5 miles per hour.x � 40 � 191.5

x 151.5

0 � x2 � 40x � 29,000

725x � 29,000 � 725x � x2 � 40x

145�5��x � 40� � 145�5�x � x�x � 40�

145

x�

145

x � 40�

1

5

t �145

x � 40�

1

5.

t �12

60�

145

x � 40

t � 145�x.x �

Time �Distance

Rate

Page 75: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.6 Other Types of Equations 141

90. Model:

Labels: Distance 1080

Original time

Original rate

Return time

Return rate

Equation:

The original rate was miles per hour.108020 � 54

t � 20 � 0 ⇒ t � 20

2t � 45 � 0 ⇒ t � �22.5

�2t � 45��t � 20� � 0

2t2 � 5t � 900 � 0

2700 � 6t2 � 15t � 0

2700

t� 6t � 15 � 0

1080 �2700

t� 6t � 15 � 1080

�1080

t� 6��t � 2.5� � 1080

�1080

t� 6

� t � 2.5

�1080

t

� t

�Rate� � �time� � �distance� 91.

r 0.04 � 4%

��1.220996�1�60 � 1��12� � r

�1.220996�1�60 � 1 �r

12

1.220996 � �1 �r

12�60

3052.49 � 2500�1 �r

12��12��5�

A � P�1 �r

n�nt

92. (a) Use

(b)

4��16

5 �1�80

� 1� � r 0.059 or 5.9%

�16

5 �1�80

� 1 �r

4

16

5� �1 �

r

4�80

32,000 � 10,000�1 �r

4�80

4��5

2�1�80

� 1� � r 0.046 or 4.6%

�5

2�1�80

� 1 �r

4

�5

2�1�80

� 1 �r

4

5

2� �1 �

r

4�80

25,000 � 10,000�1 �r

4�80

25,000 � 10,000�1 �r

4�4�20

A � P�1 �r

n�nt

. 93.

(a)

The number of medical doctors reached 816,000 lateduring the year 1999.

(b)

The model predicts the number of medical doctorswill reach 900,000 during the year 2005.

The actual number of medical doctors in 2005 isabout 800,000.

t 15.27

t � �436.03111.6 �

2

111.6�t � 436.03

463.97 � 111.6�t � 900

t 9.95

t � �352.03111.6 �

2

111.6�t � 352.03

463.97 � 111.6�t � 816

M � 463.97 � 111.6�t, 4 ≤ t ≤ 12

Page 76: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

142 Chapter 1 Equations, Inequalities, and Mathematical Modeling

94. (a) million when:

So the total voting-age population reached 200 million during 1998.

t � 8.7

2.011t � 17.55

182.45 � 3.189t � 200 � 5.2t

182.45 � 3.189t � 200�1.00 � 0.026t�

182.45 � 3.189t1.00 � 0.026t

� 200

P � 200 (b) For

The model predicts that the total voting-age populationwill reach 230 million during 2007. This value is reason-able but the model is reaching its limit since it soonbegins to rise very fast due to its asymptotic behavior.

t � 17

so 2.791t � 47.55

182.45 � 3.189t � 230 � 5.98

182.45 � 3.189t � 230�1.00 � 0.026t�

182.45 � 3.189t1.00 � 0.026t

� 230

P � 230:

95.

(a)

(b) when pounds per square inch.x 15T � 212�

T � 75.82 � 2.11x � 43.51�x, 5 ≤ x ≤ 40

96. When we have:

x � 26.25 � 26,250 passengers

5.25 � 0.2x

6.25 � 0.2x � 1

2.5 � �0.2x � 1

C � 2.5 97.

Rounding x to the nearest whole unit yields units.x 500

� 10.01x � 500.25

� 10.01x � 5.0025

0.01x � 1 � 6.0025

�0.01x � 1 � 2.45

1x � 37.55 � 40 � �0.01x � 1

98. When we have:

So the demand is 249,900 units when the price is $750.

x � 249,900

0.01x � 2499

2500 � 0.01x � 1

50 � �0.01x � 1

750 � 800 � �0.01x � 1

p � $750,

(c)

By the Quadratic Formula, we have

Since x is restricted to let pounds per square inch.

(d)

05 40

300

(14.806119, 212)

x � 14.8065 ≤ x ≤ 40,

�x 3.84787 ⇒ x 14.806

�x 16.77298 ⇒ x 281.333

0 � �2.11x � 43.51�x � 136.18

212 � 75.82 � 2.11x � 43.51�xAbsolute TemperaturePressure,

5 162.56

10 192.31

15 212.68

20 228.20

25 240.62

30 250.83

35 259.38

40 266.60

Tx

Page 77: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.6 Other Types of Equations 143

100.

(a)

(b)

when is between 170 and 175 feet.hd � 200

d � 200 when h 173 feet.

02400

300

d � �1002 � h2

(c)

(d) Solving graphically or numerically yields an approximation. An exact solution is obtained algebraically.

h 173.2 feet

h � �30,000

30,000 � h2

40,000 � 10,000 � h2

200 � �1002 � h2

101.

(a)

When

(b)

when h is between 11 and 12 inches.S � 350

S � 350, h 11.4.

0150

450

S � 8�64 � h2

(c)

(d) Solving graphically or numerically yields anapproximate solution. An exact solution is obtained algebraically.

h 11.4

h2 129.935

64 � h2 193.935

122,500 � 642�64 � h2�

�350�2 � �8�64 � h2 �2

350 � 8�64 � h2

102. Let the number of hours required for the faster person to complete the task alone. Then the number of hours needed for the slower person to complete the same task alone. In one hour, the faster person completes of the task, the slower person of the task, and together they complete of the task.

By completing the square, we have Choosing the positive value for x, we have the following times for each person:

Faster person: slower person: 9 � �65 17 hours 7 � �65 15 hours;

x � 7 ± �65.

0 � x2 � 14x � 16

8x � 16 � 8x � x2 � 2x

8�x � 2� � 8x � x�x � 2�

1x

�1

x � 2�

18

1�81��x � 2�1�x

x � 2 �x �

160 165 170 175 180 185

188.7 192.9 197.2 201.6 205.9 210.3d

h

8 9 10 11 12 13

284.3 302.6 321.9 341.8 362.5 383.6S

h

99. Model:

Labels:

Equation:

The distance between bases is approximately 90 feet.

x � ±�8128.125 ±90

x2 �16,256.25

2

2x2 � 16,256.25

x2 � x2 � �127.5�2

distance from home to 2nd � 127.5distance from 1st to 2nd � x,Distance from home to 1st � x,

�Distance fromhome to 1st �

2

� �distance from1st to 2nd �

2

� �distance fromhome to 2nd �

2

Page 78: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

144 Chapter 1 Equations, Inequalities, and Mathematical Modeling

103. Model:

Labels: Work done rate of first person time worked by first person

rate of second person time worked by second person

Equation:

(Choose the positive value for r.)

It would take approximately 23 hours and 26 hours individually.

r 23

r ����21� ± ���21�2 � 4�1���36�

2�1��

21 ± �585

2

0 � r2 � 21r � 36

12r � 36 � 12r � r2 � 3r

r�r � 3�12

r� r�r � 3�

12

r � 3� r�r � 3�

12

r�

12

r � 3� 1

� 12�1

r � 3,

� 12,�1

r,� 1,

� Portion doneby first person� � � portion done

by second person� � �workdone�

104. (a) Verbal Model: Total cost Cost underwater Distance underwater Cost overland Distance overland

Labels: Total cost:

Cost overland: $24 per foot

Distance overland in feet:

Cost underwater: $30 per foot

Distance underwater in feet:

Equation:

(b)

So, the total cost when is $1,123,424.95.

(c)

By the Quadratic Formula, we have miles or mile.

—CONTINUED—

x 0.382x 2

0 � 144x2 � 343.04x � 110.0816

256x2 � 343.04x � 114.9184 � 400x2 � 225

256x2 � 343.04x � 114.9184 � 25�16x2 � 9�

�16x � 10.72�2 � �5�16x2 � 9�2

16x � 10.72 � 5�16x2 � 9

138.72 � 128 � 16x � 5�16x2 � 9

138.72 � 16�8 � x� � 5�16x2 � 9

1,098,662.40 � 7920�16�8 � x� � 5�16x2 � 9� 1,098,662.40 � 126,720�8 � x� � 39,600�16x2 � 9

x � 3

1,123,424.95

633,600 � 489,824.95

� 126,720�5� � 39,600�153

C � 126,720�8 � 3� � 39,600�16�3�2 � 9

C � 126,720�8 � x� � 39,600�16x2 � 9

C � 24�5280�8 � x�] � 30�1320�16x2 � 9�� 1320�16x2 � 95280�x2 � �3�4�2

5280�8 � x�

C

����

Page 79: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.6 Other Types of Equations 145

104. —CONTINUED—

(d)

00 10

1,500,000 (e) The cost is a minimum when (The minimum costis $1,085,040.00.)

x � 1.

105.

v2s

R� g

v2s � gR

v2 �gRs

v ��gRs

106.

±�LCi2 � q � Q

LCi2 � q � Q2

LCi2 � Q2 � q

i2 �1

LC�Q2 � q�

i2 � �±� 1

LC�Q2 � q�2

i � ±� 1

LC�Q2 � q

107. False—See Example 7 on page 137. 108. False. has only one solution to check, 0.x � 0

109. The distance between is 13.

Both and are a distance of 13from �1, 2�.

�6,�10���4, �10�

x � 6 � 0 ⇒ x � 6

x � 4 � 0 ⇒ x � �4

�x � 4��x � 6� � 0

x2 � 2x � 24 � 0

x2 � 2x � 1 � 144 � 169

�x � 1�2 � ��12�2 � 132

��x � 1�2 � ��10 � 2�2 � 13

�1, 2� and �x, �10� 110. The distance between is 13.

Both and are a distance of 13from ��8, 0�.

�4, 5���20, 5�

x � 4 � 0 ⇒ x � 4

x � 20 � 0 ⇒ x � �20

�x � 20��x � 4� � 0

x2 � 16x � 80 � 0

x2 � 16x � 64 � 25 � 169

�x � 8�2 � 52 � 132

��x � 8�2 � �5 � 0�2 � 13

��8, 0� and �x, 5�

111. The distance between is 17.

Both and are a distance of 17from �0, 0�.

�8, �15��8, 15�

� ±15

y2 � ±�225

y2 � 225

64 � y2 � 289

�8�2 � �y�2 � 172

��8 � 0�2 � �y � 0�2 � 17

�0, 0� and �8, y� 112. The distance between is 17.

Both and are a distance of 17from ��8, 4�.

�7, 12��7, �4�

y � 4 ± 8 � �4, 12

y � 4 � ±8

�y � 4�2 � 64

225 � �y � 4�2 � 289

�15�2 � �y � 4�2 � 172

��7 � ��8��2 � �y � 4�2 � 17

��8, 4� and �7, y�

Page 80: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

146 Chapter 1 Equations, Inequalities, and Mathematical Modeling

113.

OR

Thus, or From the original equationwe know that

Some possibilities are:

or

or

or

or

or a � 14a � 4b � 14,

a � 13a � 5b � 13,

a � 12a � 6b � 12,

a � 11a � 7b � 11,

a � 10a � 8b � 10,

a � 9b � 9,

b ≥ 9.a � b.a � 18 � b

a � 18 � b

�a � b � 18

9 � a � b � 9

9 � a � b � 9

9 � 9 � a � b 114. Isolate the absolute value by subtracting x from bothsides of the equation. The expression inside the absolutevalue signs can be positive or negative, so two separateequations must be solved. Each solution must bechecked since extraneous solutions may be included.

a � b

�a � �b

9 � a � �b � 9

9 � a � ��b � 9�

115.

This formula gives the relationship between a and b. From the original equation we know that and

Choose a b value, where and then solve for a, keeping in mind that

Some possibilities are:

b � 25, a � �5

b � 24, a � 14

b � 23, a � 11

b � 22, a � 16

b � 21, a � 19

b � 20, a � 20

a ≤ 20.b ≥ 20b ≥ 20.

a ≤ 20

a � �b2 � 40b � 380

�a � b2 � 40b � 380

20 � a � b2 � 40b � 400

�20 � a � b � 20

20 � �20 � a � b 116. First isolate the radical by subtracting x from both sidesof the equation. Then, square both sides and solve theresulting equation using the Quadratic Formula. Eachsolution must be checked since extraneous solutions maybe included.

117.83x

�32x

�166x

�96x

�256x

118.

�x � 4

�x � 1�(x2 � 4�

�2x � 2 � x � 2

�x � 1��x2 � 4�

�2�x � 1� � �x � 2�

�x � 1��x � 2��x � 2�

2

x2 � 4�

1

x2 � 3x � 2�

2

�x � 2��x � 2��

1

�x � 1��x � 2�119.

��3z2 � 2z � 4

z�z � 2�

�2z � 3z2 � 6z � 2z � 4

z�z � 2�

�2z � 3z�z � 2� � 2�z � 2�

z�z � 2�

2

z � 2� �3 �

2z� �

2z � 2

� 3 �2z

Page 81: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.7 Linear Inequalities in One Variable 147

120.

�125y

x, y � 0

25y2 �xy

5�

25y2

1�

5

xy121.

x � 11

x � 11 � 0

�x � 11�2 � 0

x2 � 22x � 121 � 0

122.

x � 20 � 0 ⇒ x � 20

x � 3 � 0 ⇒ x � �3

�x � 3��x � 20� � 0

x�x � 20� � 3�x � 20� � 0

Section 1.7 Linear Inequalities in One Variable

■ You should know the properties of inequalities.

(a) Transitive: and implies

(b) Addition: and implies

(c) Adding or Subtracting a Constant: if

(d) Multiplying or Dividing a Constant: For

1. If then and

2. If then and

■ You should be able to solve absolute value inequalities.

(a) if and only if

(b) if and only if x < �a or x > a.�x� > a

�a < x < a.�x� < a

a

c>

b

c.ac > bcc < 0,

a

c<

b

c.ac < bcc > 0,

a < b,

a < b.a ± c < b ± c

a � c < b � d.c < da < b

a < c.b < ca < b

1. Interval:

(a) Inequality:

(b) The interval is bounded.

�1 ≤ x ≤ 5

��1, 5� 2. Interval:

(a) Inequality:

(b) The interval is bounded.

2 < x ≤ 10

�2, 10� 3. Interval:

(a) Inequality:

(b) The interval is unbounded.

x > 11

�11, ��

4. Interval:

(a) Inequality:

(b) The interval is unbounded.

�5 ≤ x < � or x ≥ �5

��5, �� 5. Interval:

(a) Inequality:

(b) The interval is unbounded.

x < �2

���, �2� 6. Interval:

(a) Inequality:or

(b) The interval is unbounded.

x ≤ �7�� < x ≤ 7

���, 7�

7.

Matches (b).

x < 3 8.

Matches (f).

x ≥ 5 9.

Matches (d).

�3 < x ≤ 4 10.

Matches (c).

0 ≤ x ≤ 92

Vocabulary Check

1. solution set 2. graph 3. negative

4. equivalent 5. double 6. union

Page 82: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

148 Chapter 1 Equations, Inequalities, and Mathematical Modeling

11.

Matches (e).

�x� < 3 ⇒ �3 < x < 3 12.

Matches (a).

�x� > 4 ⇒ x > 4 or x < �4

13.

(a)

Yes, is asolution.

x � 3

3 > 0

5�3� � 12 >?

0

x � 3

5x � 12 > 0

(b)

No, is not asolution.

x � �3

�27 >� 0

5��3� � 12 >?

0

x � �3 (c)

Yes, is a solution.x �52

12 > 0

5�52� � 12 >

?0

x �52 (d)

No, is not a solution.

x �32

�92 >� 0

5�32� � 12 >

?0

x �32

14.

(a)

No, is nota solution.

x � 0

1 <� �3

2�0� � 1 <?

�3

x � 0

2x � 1 < �3

(b)

No, is nota solution.

x � �14

12 <� �3

2��14� � 1 <

?�3

x � �14 (c)

Yes, is asolution.

x � �4

�7 < �3

2��4� � 1 <?

�3

x � �4 (d)

No, is nota solution.

x � �32

�2 <� �3

2��32� � 1 <

?�3

x � �32

15.

(a)

Yes, is asolution.

x � 4

0 <12

< 2

0 <? 4 � 2

4<?

2

x � 4

0 <x � 2

4< 2

(b)

No, is not asolution.

x � 10

0 < 2 <� 2

0 <? 10 � 2

4<?

2

x � 10 (c)

No, is not a solution.

x � 0

0 <� �12

< 2

0 <? 0 � 2

4<?

2

x � 0 (d)

Yes, is a solution.x �72

0 <38

< 2

0 <? �7�2� � 2

4<?

2

x �7

2

16.

(a)

No, is not asolution.

x � 0

�1 <32

≤� 1

�1 <? 3 � 0

2≤? 1

x � 0

�1 <3 � x

2≤ 1

(b)

No, is not asolution.

x � �5

�1 < 4 ≤ 1

�1 <? 3 � 5

2≤? 1

x � �5 (c)

Yes, is a solution.x � 1

�1 < 1 ≤ 1

�1 <? 3 � 1

2≤? 1

x � 1 (d)

No, is not a solution.

x � 5

�1 <?

�1 ≤ 1

�1 <? 3 � 5

2≤? 1

x � 5

17.

(a)

Yes, is asolution.

x � 13

3 ≥ 3

�13 � 10� ≥?

3

x � 13

�x � 10� ≥ 3

(b)

Yes, is a solution.

x � �1

11 ≥ 3

��1 � 10� ≥?

3

x � �1 (c)

Yes, is a solution.

x � 14

4 ≥ 3

�14 � 10� ≥?

3

x � 14 (d)

No, is not a solution.

x � 9

1 ≥� 3

�9 � 10� ≥?

3

x � 9

Page 83: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.7 Linear Inequalities in One Variable 149

18.

(a)

No, is nota solution.

x � �6

15 <� 15

�2��6� � 3� <?

15

x � �6

�2x � 3� < 15

(b)

Yes, is asolution.

x � 0

3 < 15

�2�0� � 3� <?

15

x � 0 (c)

No, is nota solution.

x � 12

21 <� 15

�2�12� � 3� <?

15

x � 12 (d)

Yes, is asolution.

x � 7

11 < 15

�2�7� � 3� <?

15

x � 7

19.

1 2 3 4 5

x

x < 3

14�4x� < 14�12�

4x < 12 20.

−2−6 −5 −4 −3

x

x < �4

10x < �40 21.

2 310−1

x

−2

32

x < 32

�12��2x� < ��1

2���3�

�2x > �3

22.

2− 5

−1−2−3−4

x

x < �156 or x < �

52

�6x > 15 23.

10 11 12 13 14

x

x ≥ 12

x � 5 ≥ 7 24.

3 4 5 6 7

x

x ≤ 5

x � 7 ≤ 12

25.

0 1 2 3 4

x

x > 2

�2x < �4

2x � 7 < 3 � 4x 26.

−1 0 1

1

2

2x

x ≥ 12

2x ≥ 1

3x � 1 ≥ 2 � x 27.

−2 −1 10 2

x

27

x ≥ 27

7x ≥ 2

2x � 1 ≥ 1 � 5x

28.

x

−1−2−3−4−5

x ≥ �3

�2x ≤ 6

6x � 4 ≤ 2 � 8x 29.

43 5 6 7

x

x < 5

4 � 2x < 9 � 3x

4 � 2x < 3�3 � x� 30.

−2

2

−1

−1

0 1x

x < �12

2x < �1

4x � 4 < 2x � 3

4�x � 1� < 2x � 3

31.

432 5 6

x

x ≥ 4

�14x ≤ �1

34x � 6 ≤ x � 7 32.

5 6 7 8 9

x

x < 7

�5x > �35

21 � 2x > 7x � 14

3 �27x > x � 2 33.

210 3 4

x

x ≥ 2

4x �12 ≥ 3x �

52

12�8x � 1� ≥ 3x �52

Page 84: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

150 Chapter 1 Equations, Inequalities, and Mathematical Modeling

34.

0 21 3

x

−1

16

x > 16

�12x < �2

36x � 4 < 48x � 6

9x � 1 < 34�16x � 2� 35.

−2−3−4−5−6

x

x ≥ �4

3.6x ≥ �14.4

3.6x � 11 ≥ �3.4 36.

15 16 17 18

x

14

x > 16

�1.3x < �20.8

15.6 � 1.3x < �5.2

37.

30 1 2−1

x

�1 < x < 3

�2 < 2x < 6

�1 < 2x � 3 < 9 38.

10−1−2−3−4−5−6

x

�6 < x ≤ 1

�3 ≤ �3x < 18

�8 ≤ �3x � 5 < 13

�8 ≤ ��3x � 5� < 13 39.

−2−4−6

0 2 64 8

x29

215

1�92 < x < 15

2

1�9 < 2x < 15

�12 < 2x � 3 < 12

1�4 <2x � 3

3< 4

40.

−2−4

−3 7

86420

x

�3 ≤ x < 7

0 ≤ x � 3 < 10

0 ≤x � 3

2< 5 41.

0 1−1

x

−43

41

� 34

< x < �14

�14

> x > �34

34

> x � 1 >14

42.

1 3 5 7 9 11

x

3 < x < 9

�9 < �x < �3

�3 < 6 � x < 3

�1 < 2 �x3

< 1

43.

121110 13 14

x

13.510.5

10.5 ≤ x ≤ 13.5

4.2 ≤ 0.4x ≤ 5.4

3.2 ≤ 0.4x � 1 ≤ 4.4 44.

There is no solution.

2 > x > 10

3 > 1.5x > 15

9 > 1.5x � 6 > 21

4.5 >1.5x � 6

2> 10.5 45.

−2−4−6 0 2 4 6

x

�6 < x < 6

�x� < 6

46.

−6 −4 −2 0 2 4 6

x

x < �4 or x > 4

�x� > 4 47.

−2−3 −1 0 1 2 3

x

x < �2 x > 2

x2

< �1 or x2

> 1

�x2� > 1 48.

−20 −10 0 10

x

20

x < �15 x > 15

x5

< �3 or x5

> 3

�x5� > 3

49.

No solution. The absolute value of a number cannot beless than a negative number.

�x � 5� < � 1 50. There is no solution because the absolute value of a num-ber cannot be less than a negative number.

Page 85: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.7 Linear Inequalities in One Variable 151

51.

201510 25 30

x

14 26

14 ≤ x ≤ 26

�6 ≤ x � 20 ≤ 6

�x � 20� ≤ 6 52.

All real numbers x

1 2 30−3 −2 −1

x

x ≤ 8

�x ≥ �8

x ≥ 8 �x � 8 ≥ 0

x � 8 ≥ 0 or ��x � 8� ≥ 0

�x � 8� ≥ 0

53.

or

−2 −1 10 2 3 4

x

32

x ≤ �32 x ≥ 3

�4x ≥ 6 �4x ≤ �12

3 � 4x ≥ 9 3 � 4x ≤ �9

�3 � 4x� ≥ 9 54.

−1−2 10 2 3

x

�2 < x < 3

3 > x > �2

�6 < �2x < 4

�5 < 1 � 2x < 5

�1 � 2x� < 5

55.

or

0−5−10−15 5 10 15

x

11

x ≥ 11 x ≤ �5

x � 3 ≥ 8 x � 3 ≤ �8

x � 3

2≥ 4

x � 3

2≤ �4

�x � 32 � ≥ 4 56.

−1 0 1 2 3 4

x

0 < x < 3

3 > x > 0

�2 < �2x

3< 0

�1 < 1 �2x

3< 1

�1 �2x

3 � < 1

57.

x

3 654

4 < x < 5

5 > x > 4

�10 < �2x < �8

�1 < 9 � 2x < 1

�9 � 2x� < 1

�9 � 2x� � 2 < �1 58.

−35 −28 −21 −14 70−7

x

x < �28 x > 0

x � 14 < �14 or x � 14 > 14

�x � 14� > 14

�x � 14� � 3 > 17

59.

or

−12

− −

−16 −4−8

x

112

292

x ≥ �112 x ≤ �

292

x � 10 ≥ 92 x � 10 ≤ �

92

�x � 10� ≥ 92

2�x � 10� ≥ 9 60.

15 ≤ x ≤ 75

75 ≥ x ≥ 15

�7 ≤ �5x ≤ �1

�3 ≤ 4 � 5x ≤ 3

�4 � 5x� ≤ 32

755

1

1

0

x

3�4 � 5x� ≤ 9

Page 86: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

152 Chapter 1 Equations, Inequalities, and Mathematical Modeling

61.

−10 10

−10

10

6x > 2

6x > 12 62.

−10 10

−10

10

x ≤ 2

3x ≤ 6

3x � 1 ≤ 5 63.

−10 10

−10

10

5 � 2x ≤ 2

5�2x ≥ �4

5 � 2x ≥ 1

64.

−10 10

−10

10

x < 2

2x < 4

3x � 3 < x � 7

3�x � 1� < x � 7 65.

10

−10

−10 24

�6 ≤ x ≤ 22

�14 ≤ x � 8 ≤ 14

�x � 8� ≤ 14 66.

−15 9

−10

10

x < �11 x > 2

2x < �22 2x > 4

2x � 9 < �13 or 2x � 9 > 13

�2x � 9� > 13

67.

or

−15 1

−10

10

x ≥ �12 x ≤ �

272

x � 7 ≥ 132 x � 7 ≤ �

132

�x � 7� ≥ 132

2�x � 7� ≥ 13 68.

−10 10

−10

10

�7 ≤ x ≤ 5

�6 ≤ x � 1 ≤ 6

�x � 1� ≤ 6

12�x � 1� ≤ 3

69.

(a)

(b)

2 � 3x ≤ 32

� 32x ≤ 3

2x � 3 ≤ 0

2 � 3y ≤ 0

2 � 3x ≥ 2

� 32x ≥ 4

2x � 3 ≥ 1

2 � 3y ≥ 1−4 8

−5

3y � 2x � 3 70.

(a)

(b)

x ≥ �32

23x ≥ �1

23x � 1 ≥ 0

y ≥ 0

x ≤ 6

23x ≤ 4

23x � 1 ≤ 5

y ≤ 5

−5 7

−2

6y �23x � 1

Page 87: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.7 Linear Inequalities in One Variable 153

71.

(a)

(b)

x ≤ 4

�12x ≥ �2

�12x � 2 ≥ 0

y ≥ 0

4 ≥ x ≥ �2

�2 ≤ �12x ≤ 1

0 ≤ �12x � 2 ≤ 3

0 ≤ y ≤ 3

−6 6

−2

6y � �12x � 2 72.

(a)

(b)

x ≥ 83

�3x ≤ �8

�3x � 8 ≤ 0

y ≤ 0

53 ≤ x ≤ 3

3 ≥ x ≥ 53

�9 ≤ �3x ≤ �5

�1 ≤ �3x � 8 ≤ 3

�1 ≤ y ≤ 3

−3

9−9

9y � �3x � 8

73.

(a)

(b)

or

or x ≥ 7 x ≤ �1

x � 3 ≥ 4 x � 3 ≤ �4

�x � 3� ≥ 4

y ≥ 4

1 ≤ x ≤ 5

�2 ≤ x � 3 ≤ 2

�x � 3� ≤ 2

y ≤ 2

−5 10

−2

8y � �x � 3� 74.

(a)

(b)

x ≤ �4 x ≥ 0

12x ≤ �2 12x ≥ 0

12x � 1 ≤ �1 or 12x � 1 ≥ 1

�12x � 1� ≥ 1

y ≥ 1

�10 ≤ x ≤ 6

�5 ≤ 12x ≤ 3

�4 ≤ 12x � 1 ≤ 4

�12x � 1� ≤ 4

y ≤ 4

−5 1

−1

3y � �12x � 1�

75.

�5, ��

x ≥ 5

x � 5 ≥ 0 76.

�10, ��

x ≥ 10

x � 10 ≥ 0

�x � 10 77.

��3, ��

x ≥ �3

x � 3 ≥ 0 78.

���, 3�

3 ≥ x

3 � x ≥ 0

�3 � x

79.

���, 72� x ≤ 7

2

�2x ≥ �7

7 � 2x ≥ 0 80.

��52, ��

x ≥ �52

6x ≥ �15

6x � 15 ≥ 0

4�6x � 15 81.

All real numbers within8 units of 10.

�x � 10� < 8 82.

All real numbers morethan 4 units from 8

�x � 8� > 4

83. The midpoint of the interval is 0. The interval represents all realnumbers x no more than 3 unitsfrom 0.

�x� ≤ 3

�x � 0� ≤ 3

��3, 3� 84. The graph shows all real numbersmore than 3 units from 0.

�x� > 3

�x � 0� > 3

85. The graph shows all real numbersat least 3 units from 7.

�x � 7� ≥ 3

Page 88: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

154 Chapter 1 Equations, Inequalities, and Mathematical Modeling

86. The graph shows all real numbersno more than 4 units from

�x � 1� ≤ 4

�1.87. All real numbers within 10 units

of 12

�x � 12� < 10

88. All real numbers at least 5 unitsfrom 8

�x � 8� ≥ 5

89. All real numbers more than 4 units from

�x � 3� > 4

�x � ��3�� > 4

�3 90. All real numbers no more than 7 units from

�x � 6� ≤ 7

�6

91. Let

Type A account charges:

Type B account charges:

If you write more than six checks a month, then thecharges for the type A account are less than the chargesfor the type B account.

6 < x

1.50 < 0.25x

6.00 � 0.25x < 4.50 � 0.50x

4.50 � 0.50x

6.00 � 0.25x

x � the number of checks written in a month. 92.

You must make more than 42,857 copies to justify buyingthe copier.

42,857 < x

3000 < 0.07x

3000 � 0.03x ≤ 0.1x

93.

r > 3.125%

r > 0.03125

2r > 0.0625

1 � 2r > 1.0625

1000�1 � r�2�� > 1062.50 94.

The rate must be more than 5%.

0.05 < r

75 < 1500r

825 < 750 � 1500r

825 < 750�1 � 2r�

825 < 750�1 � r�2��

95.

120.95x ≥ 36 units

120.95x > 35.7995

120.95x > 750

115.95x > 95x � 750

115.95R > C 96.

Because the number of units x must be an integer, theproduct will return a profit when at least 16,394 units are sold.

x > 16,393.44262

9.15 > 150,000

24.55x > 15.4x � 150,000

97. Let

Revenue:

Cost:

Profit:

In whole dozens, 134 ≤ x ≤ 234.

13313 ≤ x ≤ 2331

3

200 ≤ 1.50x ≤ 350

50 ≤ 1.50x � 150 ≤ 200

50 ≤ P ≤ 200

� 1.50x � 150

� 2.95x � �150 � 1.45x�

P � R � C

C � 150 � 1.45x

R � 2.95x

x � daily sales level (in dozens) of doughnuts. 98. The goal is to lose pounds. At poundsper week, it will take 24 weeks.

� 24

� 12 2

36 � 112 � 36

23

112164 � 128 � 36

Page 89: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

(e) An athlete’s weight is not a particularly good indicator of the athlete’s maximum bench pressweight. Other factors, such as muscle tone and exercise habits, influence maximum bench pressweight.

140140 260

300

Section 1.7 Linear Inequalities in One Variable 155

99. (a)

075 150

5

y � 0.067x � 5.638 (b) From the graph we see that when Algebraically we have:

IQ scores are not a good predictor of GPAs. Other factorsinclude study habits, class attendance, and attitude.

x ≥ 129

8.638 ≤ 0.067x

3 ≤ 0.067x � 5.638

x ≥ 129.y ≥ 3

100. (a) and (b)

(c) One estimate is pounds.

(d)

x ≥ 181.5385 � 181.54 pounds

1.3x ≥ 236

1.3x � 36 ≥ 200

x ≥ 181

x 165 184 150 210 196 240

y 170 185 200 255 205 295

179 203 159 237 219 2761.3x � 36

x 202 170 185 190 230 160

y 190 175 195 185 250 155

227 185 205 211 263 1721.3x � 36

101.

(a)

Rounding to the nearest year, The average salary was at least $32,000 but not morethan $42,000 between 1991 and 2000.

(b)

According to the model, the average salary willexceed $48,000 in 2006.

t > 16

1.05t > 17

1.05t � 31 > 48

1 ≤ t ≤ 10.

0.95 ≤ t ≤ 10.48

1 ≤ 1.05t ≤ 11

32 ≤ 1.05t � 31 ≤ 42

S � 1.05t � 31.0, 0 ≤ t ≤ 12 102. (a)

The number of eggs produced was between 70 and80 billion from late 1991 until late 1997.

(b) when

So the annual egg production will exceed 95 billionin 2007.

t ≥ 16.95.E � 1.64t � 67.2 ≥ 95

1.7 ≤ t ≤ 7.8

2.8 ≤ 1.64t ≤ 12.8

70 ≤ 1.64t � 67.2 ≤ 80

103.

Since we have

106.864 ≤ area ≤ 109.464.

�10.3375�2 ≤ area ≤ �10.4625�2

A � s2,

10.3375 ≤ s ≤ 10.4625

�0.0625 ≤ s � 10.4 ≤ 0.0625

� 116 ≤ s � 10.4 ≤ 1

16

�s � 10.4� ≤ 116 104.

The interval containing the possible side lengths s in centimeters of the square is so the inter-val containing the possible areas in square centimeters is

or �573.6025, 597.8025�.�23.952, 24.452�,

�23.95, 24.45�,

23.95 ≤ s ≤ 24.45

24.2 � 0.25 ≤ s ≤ 24.2 � 0.25

Page 90: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

156 Chapter 1 Equations, Inequalities, and Mathematical Modeling

105.

You might have been undercharged or overcharged by $0.19.

110�$1.89� � $0.19

14.9 ≤ x ≤ 15.1 gallons

� 110 ≤ x � 15 ≤ 1

10

�x � 15� ≤ 110 106. 1 oz lb, so oz lb

so you could have been under- or overcharged as much as $0.47.14.99 � 1

32 � 0.4684375,

�132

12�

116

107.

Two-thirds of the workers could perform the task in thetime interval between 13.7 minutes and 17.5 minutes.

13.7 < t < 17.5

�1.9 < t � 15.6 < 1.91615141312 17 18 19

t

17.513.7 �1 <t � 15.6

1.9 < 1

�t � 15.6

1.9 � < 1 108.

6968676665 70 71 72h

71.265.8

65.8 inches ≤ h ≤ 71.2 inches

�2.7 ≤ h � 68.5 ≤ 2.7

�1 ≤h � 68.5

2.7≤ 1

�h � 68.5

2.7 � ≤ 1

109.

The minimum relative humidity is 20 and the maximumis 80.

20 ≤ h ≤ 80

�30 ≤ h � 50 ≤ 30

�h � 50� ≤ 30 110. (a) Estimate from the graph: when the plate thickness is2 millimeters, the frequency is approximately 330vibrations per second.

(b) Estimate from the graph: when the frequency is 600,the plate thickness is approximately 3.6 millimeters.

(c) Estimate from the graph: when the frequency isbetween 200 and 400 vibrations per second, the platethickness is between 1.2 and 2.4 millimeter.

(d) Estimate from the graph: when the plate thickness is less than 3 millimeters, the frequency is less than500 vibrations per second.

114. must be greater than or equal to zero.

Let then and This is true when One set of values is

(Note: This solution is not unique. Any positive multiple of these values will also work, such as or a � 3, b � c � 15.�b � c � 10a � 2,

c � 5.b � 5,a � 1,b � c � 5.b � c � 10.b � c � 0a � 1,

b � c ≤ ax ≤ b � c

�c ≤ ax � b ≤ c

�ax � b� ≤ c ⇒ c

111. False. If c is negative, thenac ≥ bc.

112. False. If thenand �x ≥ �8.10 ≥ �x

�10 ≤ x ≤ 8, 113. Matches (b).

or

x ≥ a � 2

x � a ≥ 2

x ≤ a � 2

x � a ≤ �2

�x � a� ≥ 2

115. and

Midpoint: �4 � 12

, 2 � 12

2 � �32

, 7d � ��1 � ��4��2 � �12 � 2�2 � �52 � 102 � �125 � 5�5

�1, 12���4, 2�

Page 91: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.8 Other Types of Inequalities 157

116.

Midpoint: �1 � 102

, �2 � 3

2 � � �112

, 12�

d � ��1 � 10�2 � ��2 � 3�2 � ���9�2 � ��5�2 � �81 � 25 � �106

117. and

Midpoint: �3 � ��5�2

, 6 � ��8�

2 � � ��1, �1�

d � ���5 � 3�2 � ��8 � 6�2 � ���8�2 � ��14�2 � �260 � 2�65

��5, �8��3, 6�

118.

Midpoint: �0 � 62

, �3 � 9

2 � � ��62

, 62� � ��3, 3�

d � ��0 � ��6��2 � ��3 � 9�2 � �62 � ��12�2 � �36 � 144 � �180 � 6�5

119.

x � 10

6x � 60

6x � 24 � 36

�12 � 6x � 12 � 36

�6�2 � x� � 12 � 36 120.

x �132

�2x � �13

4x � 28 � 9 � 6x � 6

4�x � 7� � 9 � �6��x � 1�

121.

or

x � �12 x �

17

2x � �1 7x � 1

2x � 1 � 0 7x � 1 � 0

�7x � 1��2x � 1� � 0

14x2 � 5x � 1 � 0 122.

x2 � 4 � 0 ⇒ x � ±2

x � 5 � 0 ⇒ x � �5

�x � 5��x2 � 4� � 0

x2�x � 5� � 4�x � 5� � 0

x3 � 5x2 � 4x � 20 � 0

123. ��3, 10� 125. Answers will vary.124. If then the point couldbe located in either Quadrant I or II.

�x, y�y > 0,

Section 1.8 Other Types of Inequalities

■ You should be able to solve inequalities.

(a) Find the critical number.

1. Values that make the expression zero

2. Values that make the expression undefined

(b) Test one value in each test interval on the real number line resulting from the critical numbers.

(c) Determine the solution intervals.

Vocabulary Check

1. critical; test intervals 2. zeroes; undefined values 3. P � R � C

Page 92: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

158 Chapter 1 Equations, Inequalities, and Mathematical Modeling

1.

(a)

No, is not asolution.

x � 3

6 <� 0

�3�2 � 3 <?

0

x � 3

x2 � 3 < 0

(b)

Yes, is a solution.x � 0

�3 < 0

�0�2 � 3 <?

0

x � 0 (c)

Yes, is a solution.x �32

�34 < 0

�32�2

� 3 <?

0

x �32 (d)

No, is not asolution.

x � �5

22 <� 0

��5�2 � 3 <?

0

x � �5

2.

(a)

Yes, is a solution.

x � 5

8 ≥ 0

�5�2 � �5� � 12 ≥?

0

x � 5

x2 � x � 12 ≥ 0

(b)

No, is not a solution.

x � 0

�12 ≥� 0

�0�2 � 0 � 12 ≥?

0

x � 0 (c)

Yes, is a solution.x � �4

8 ≥ 0

16 � 4 � 12 ≥?

0

��4�2 � ��4� � 12 ≥?

0

x � �4 (d)

Yes, is a solution.x � �3

0 ≥ 0

9 � 3 � 12 ≥?

0

��3�2 � ��3� � 12 ≥?

0

x � �3

3.

(a)

Yes, is asolution.

x � 5

7 ≥ 3

5 � 25 � 4

≥?

3

x � 5

x � 2x � 4

≥ 3

(b)

is undefined.

No, is not asolution.

x � 4

60

4 � 24 � 4

≥?

3

x � 4 (c)

No, is not asolution.

x � �92

5

17≥� 3

92 � 2

�92 � 4

≥?

3

x � �9

2(d)

Yes, is a solution.x �92

13 ≥ 3

92 � 292 � 4

≥?

3

x �9

2

4.

(a)

No, is nota solution.

x � �2

12

8<� 1

3��2�2

��2�2 � 4<?

1

x � �2

3x2

x2 � 4< 1

(b)

Yes, is a solution.

x � �1

3

5< 1

3��1�2

��1�2 � 4<?

1

x � �1 (c)

Yes, is a solution.

x � 0

0 < 1

3�0�2

�0�2 � 4<?

1

x � 0 (d)

No, is not a solution.

x � 3

27

13<� 1

3�3�2

�3�2 � 4<?

1

x � 3

5.

Critical numbers: x � �32

, x � 2

x � 2 � 0 ⇒ x � 2

2x � 3 � 0 ⇒ x � �32

2x2 � x � 6 � �2x � 3��x � 2� 6.

The critical numbers are 0 and 25

9.

9x � 25 � 0 ⇒ x �25

9

x2 � 0 ⇒ x � 0

x2�9x � 25� � 0

9x3 � 25x2 � 0 7.

Critical numbers: x �72

, x � 5

⇒ x � 5 x � 5 � 0

⇒ x �72

2x � 7 � 0

�2x � 7x � 5

2 �3

x � 5�

2�x � 5� � 3x � 5

Page 93: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.8 Other Types of Inequalities 159

8.

The critical numbers are �2, �1, 1, 4.

x � 1 � 0 ⇒ x � 1

x � 2 � 0 ⇒ x � �2

�x � 2��x � 1� � 0

x � 1 � 0 ⇒ x � �1

x � 4 � 0 ⇒ x � 4

�x � 4��x � 1� � 0

��x � 4��x � 1��x � 2��x � 1�

�x2 � x � 2x � 4

�x � 2��x � 1�

x

x � 2�

2

x � 1�

x�x � 1� � 2�x � 2��x � 2��x � 1�

9.

Critical numbers:

Test intervals:

Test: Is

Interval x-Value Value of Conclusion

Positive

Negative

Positive

Solution set:

−1−2−3 0 1 2 3

x

��3, 3�

16 � 9 � 7x � 4�3, ��

0 � 9 � �9x � 0��3, 3�

16 � 9 � 7x � �4���, �3�

x2 � 9

�x � 3��x � 3� ≤ 0?

���, �3�, ��3, 3�, �3, ��

x � ±3

�x � 3��x � 3� ≤ 0

x2 � 9 ≤ 0

x2 ≤ 9

10.

Critical numbers:

Test intervals:

Solution interval:

x

−8 −6 −4 −2 0 2 4 6 8

��6, 6�

�6, �� ⇒ �x � 6��x � 6� > 0

��6, 6� ⇒ �x � 6��x � 6� < 0

���, �6� ⇒ �x � 6��x � 6� > 0

x � �6, x � 6

�x � 6��x � 6� < 0

x2 � 36 < 0

x2 < 36 11.

Critical numbers:

Test intervals:

Test: Is

Interval x-Value Value of Conclusion

Positive

Negative

Positive

Solution set:

−8 −4−6

−7

0 2

3

4

x

−2

��7, 3�

�12��2� � 24x � 5�3, ��

�7���3� � �21x � 0��7, 3�

��3���13� � 39x � �10���, �7�

�x � 7��x � 3�

�x � 7��x � 3� < 0?

���, �7�, ��7, 3�, �3, ��

x � �7, x � 3

�x � 7��x � 3� < 0

x2 � 4x � 21 < 0

x2 � 4x � 4 < 25

�x � 2�2 < 25

12.

Critical numbers:

Test intervals:

Solution intervals:

31 2 54

x

���, 2� � �4, ��

�4, �� ⇒ �x � 2��x � 4� > 0

�2, 4� ⇒ �x � 2��x � 4� < 0

���, 2� ⇒ �x � 2��x � 4� > 0

x � 2, x � 4

�x � 2��x � 4� ≥ 0

x2 � 6x � 8 ≥ 0

�x � 3�2 ≥ 1

Page 94: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

160 Chapter 1 Equations, Inequalities, and Mathematical Modeling

13.

Critical numbers:

Test intervals:

Test: Is

Interval x-Value Value of Conclusion

Positive

Negative

Positive

Solution set:

x

210−1−2−3−4−5−6

���, �5� � �1, ��

�7��1� � 7x � 2�1, ��

�5���1� � �5x � 0��5, 1�

��1���7� � 7x � �6���, �5�

�x � 5��x � 1�

�x � 5��x � 1� ≥ 0?

���, �5�, ��5, 1�, �1, ��

x � �5, x � 1

�x � 5��x � 1� ≥ 0

x2 � 4x � 5 ≥ 0

x2 � 4x � 4 ≥ 9 14.

Critical numbers:

Test intervals:

Solution interval:

−2

−1

2 40 6

7

8

x

��1, 7�

�7, �� ⇒ �x � 1��x � 7� > 0

��1, 7� ⇒ �x � 1��x � 7� < 0

���, �1� ⇒ �x � 1��x � 7� > 0

x � �1, x � 7

�x � 1��x � 7� < 0

x2 � 6x � 7 < 0

x2 � 6x � 9 < 16

15.

Critical numbers:

Test intervals:

Test: Is

Interval x-Value Value of Conclusion

Positive

Negative

Positive

Solution set:

0−1−2−3 1 2

x

��3, 2�

�6��1� � 6x � 3�2, ��

�3���2� � �6x � 0��3, 2�

��1���6� � 6x � �4���, �3�

�x � 3��x � 2�

�x � 3��x � 2� < 0?

���, �3�, ��3, 2�, �2, ��

x � �3, x � 2

�x � 3��x � 2� < 0

x2 � x � 6 < 0

x2 � x < 6 16.

Critical numbers:

Test intervals:

Solution intervals:

−1−2−3−4 0 1 2

x

���, �3� � �1, ��

�1, �� ⇒ �x � 3��x � 1� > 0

��3, 1� ⇒ �x � 3��x � 1� < 0

���, �3� ⇒ �x � 3��x � 1� > 0

x � �3, x � 1

�x � 3��x � 1� > 0

x2 � 2x � 3 > 0

x2 � 2x > 3

17.

Critical numbers:

Test intervals:

Test: Is

Interval x-Value Value of Conclusion

Positive

Negative

Positive

Solution set:

10−1−2−3

x

��3, 1�

�5��1� � 5x � 2�1, ��

�3���1� � �3x � 0��3, 1�

��1���5� � 5x � �4���, �3�

�x � 3��x � 1�

�x � 3��x � 1� < 0?

���, �3�, ��3, 1�, �1, ��

x � �3, x � 1

�x � 3��x � 1� < 0

x2 � 2x � 3 < 0 18.

Critical numbers:

Test intervals:

Solution intervals:

0−2−4 2 4 6 8

x2 − 5 2 + 5

���, 2 � �5� � �2 � �5, ���2 � �5, �� ⇒ x2 � 4x � 1 > 0

�2 � �5, 2 � �5 � ⇒ x2 � 4x � 1 < 0

���, 2 � �5 � ⇒ x2 � 4x � 1 > 0

x � 2 � �5, x � 2 � �5

x �4 ± �16 � 4

2� 2 ± �5

x2 � 4x � 1 > 0

Page 95: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.8 Other Types of Inequalities 161

19.

Complete the square.

Critical numbers:

Test intervals:

Test: Is

Interval x-Value Value of Conclusion

Positive

Negative

Positive

Solution set: ��� < �4 � �21� � ��4 � �21, �� 20

x

−8 −6 −4 −2−10

−4 + 21−4 − 214 � 16 � 5 � 15x � 2��4 � �21, ��0 � 0 � 5 � �5x � 0��4 � �21, �4 � �21 �100 � 80 � 5 � 15x � �10���, �4 � �21 �

x2 � 8x � 5

x2 � 8x � 5 ≥ 0?

��4 � �21, ����4 � �21, �4 � �21�,���, �4 � �21�,x � �4 ± �21

x � �4 ± �21

x � 4 � ±�21

�x � 4�2 � 21

x2 � 8x � 16 � 5 � 16

x2 � 8x � 5 � 0

x2 � 8x � 5 ≥ 0

20.

Critical numbers:

Test intervals:

Solution interval:

x

543210−1−2

32 2

39− 32 2

39+

���, 32

��39

2 � � 32

��39

2, ��

�32

��39

2, �� ⇒ �2x2 � 6x � 15 < 0

�32

��39

2,

32

��39

2 � ⇒ �2x2 � 6x � 15 > 0

���, 32

��39

2 � ⇒ �2x2 � 6x � 15 < 0

x �32

��39

2, x �

32

��39

2

�6 ± 2�39

4�

32

±�39

2

x ����6� ± ���6�2 � 4�2���15�

2�2� �6 ± �156

4

2x2 � 6x � 15 ≥ 0

�2x2 � 6x � 15 ≤ 0 21.

Critical numbers:

Test intervals:

Test: Is ?

Interval x-Value Value of Conclusion

Negative

Positive

Negative

Positive

Solution set:

0 1 2 43

x

−1

��1, 1� � �3, ��

�5��3��1� � 15x � 4�3, ��

�3��1���1� � �3x � 2�1, 3�

�1���1���3� � 3x � 0��1, 1�

��1���3���5� � �15x � �2���, �1�

�x � 1��x � 1��x � 3�

�x � 1��x � 1��x � 3� > 0

���, �1�, ��1, 1�, �1, 3�, �3, ��

x � ±1, x � 3

�x � 1��x � 1��x � 3� > 0

�x2 � 1��x � 3� > 0

x2�x � 3� � 1�x � 3� > 0

x3 � 3x2 � x � 3 > 0

22.

Critical numbers:

Test intervals: Solution interval:

�2, �� ⇒ x3 � 2x2 � 4x � 8 > 00 1 2 3 4

x��2, 2� ⇒ x3 � 2x2 � 4x � 8 < 0

���, 2����, �2� ⇒ x3 � 2x2 � 4x � 8 < 0

x � �2, x � 2

�x � 2��x2 � 4� ≤ 0

x2�x � 2� � 4�x � 2� ≤ 0

x3 � 2x2 � 4x � 8 ≤ 0

Page 96: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

162 Chapter 1 Equations, Inequalities, and Mathematical Modeling

23.

Critical numbers:

Test intervals:

Test: Is

Interval x-Value Value of Conclusion

Negative

Positive

Negative

Positive

Solution set: ��3, 2� � �3, ��

x

−4 −3 −2 −1 0 1 2 3 4 5

�2��7��1� � 14x � 4�3, ��

�0.5��5.5���0.5� � �1.375x � 2.5�2, 3�

��2��3���3� � 18x � 0��3, 2�

��6���1���7� � �42x � �4���, �3�

�x � 2��x � 3��x � 3�

�x � 2��x � 3��x � 3� ≥ 0?

���, �3�, ��3, 2�, �2, 3�, �3, ��

x � 2, x � ±3

�x � 2��x � 3��x � 3� ≥ 0

�x � 2��x2 � 9� ≥ 0

x2�x � 2� � 9�x � 2� ≥ 0

x3 � 2x2 � 9x � 18 ≥ 0

x3 � 2x2 � 9x � 2 ≥ �20

24.

Critical numbers:

Test intervals:

Solution interval: ��132 , �2�, �2, ��

�2, �� ⇒ 2x3 � 13x2 � 8x � 52 > 0

��2, 2� ⇒ 2x3 � 13x2 � 8x � 52 < 0

��132 , �2� ⇒ 2x3 � 13x2 � 8x � 52 > 0

4

x

−2−4−6−8 20

132

−���, �132 � ⇒ 2x3 � 13x2 � 8x � 52 < 0

x � �132 , x � �2, x � 2

�2x � 13��x2 � 4� ≥ 0

x2�2x � 13� � 4�2x � 13� ≥ 0

2x3 � 13x2 � 8x � 52 ≥ 0

2x3 � 13x2 � 8x � 46 ≥ 6

25.

Critical number:

Test intervals:

Test: Is

Interval x-Value Value of Conclusion

Positive

Positive

Solution set:

210−1

x

−2

12

x �12

�1�2 � 1x � 1�12

, ��

��1�2 � 1x � 0���, 12�

�2x � 1�2

�2x � 1�2 ≤ 0?

���, 12�, �1

2, ��

x �12

�2x � 1�2 ≤ 0

4x2 � 4x � 1 ≤ 0 26.

The critical numbers are imaginary:

So the set of real numbers is the solution set.

1 2 30−3 −2 −1

x

�32

±i�23

2

x2 � 3x � 8 > 0

Page 97: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.8 Other Types of Inequalities 163

27.

Critical numbers:

Test intervals:

Test: Is

By testing an x-value in each test interval in the inequality,we see that the solution set is: ���, 0� � �0, 32�

2x2�2x � 3� < 0?

���, 0�, �0, 32�, �32, ��

x � 0, x �32

2x2�2x � 3� < 0

4x3 � 6x2 < 0 28.

Critical numbers:

Test intervals:

Solution interval: �3, ��

�3, �� ⇒ 4x2�x � 3� > 0

�0, 3� ⇒ 4x2�x � 3� < 0

���, 0� ⇒ 4x2�x � 3� < 0

x � 0, x � 3

4x2�x � 3� > 0

4x3 � 12x2 > 0

29.

Critical numbers:

Test intervals:

Test: Is

By testing an x-value in each test interval in the inequality,we see that the solution set is: ��2, 0� � �2, ��

x�x � 2��x � 2� ≥ 0?

���, �2�, ��2, 0�, �0, 2�, �2, ��

x � 0, x � ±2

x�x � 2��x � 2� ≥ 0

x3 � 4x ≥ 0 30.

Critical numbers:

Test intervals:

Solution intervals: ���, 0� � �2, ��

�2, �� ⇒ x3�2 � x� < 0

�0, 2� ⇒ x3�2 � x� > 0

���, 0� ⇒ x3�2 � x� < 0

x � 0, x � 2

x3�2 � x� ≤ 0

2x3 � x4 ≤ 0

31.

Critical numbers:

Test intervals:

Test: Is

By testing an x-value in each test interval in the inequality,we see that the solution set is: ��2, ��

�x � 1�2�x � 3�3 ≥ 0?

���, �2�, ��2, 1�, �1, �)

x � 1, x � �2

�x � 1�2�x � 2�3 ≥ 0 32.

Critical numbers:

Test intervals:

Solution intervals: ���, 0� � �0, 3� or ���, 3�

�3, �� ⇒ x4�x � 3� > 0

�0, 3� ⇒ x4�x � 3� < 0

���, 0� ⇒ x4�x � 3� < 0

x � 0, x � 3

x4�x � 3� ≤ 0

33.

−5 7

−2

6

y � �x2 � 2x � 3 (a)

(b) y ≥ 3 when 0 ≤ x ≤ 2.

y ≤ 0 when x ≤ �1 or x ≥ 3.

34.

−10

−4

14

12

y �1

2x2 � 2x � 1 (a)

when 2 � �2 ≤ x ≤ 2 � �2.y ≤ 0

�4 ± �8

2� 2 ± �2

x ����4� ± ���4�2 � 4�1��2�

2�1�

x2 � 4x � 2 ≤ 0

12

x2 � 2x � 1 ≤ 0

y ≤ 0 (b)

y ≥ 7 when x ≤ �2, x ≥ 6.

�x � 6��x � 2� ≥ 0

x2 � 4x � 12 ≥ 0

1

2x2 � 2x � 1 ≥ 7

y ≥ 7

Page 98: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

164 Chapter 1 Equations, Inequalities, and Mathematical Modeling

35.

−12 12

−8

8

y �18x3 �

12x (a)

(b) y ≤ 6 when x ≤ 4.

y ≥ 0 when �2 ≤ x ≤ 0, 2 ≤ x < �.

36.

−12 12

−24

48

y � x3 � x2 � 16x � 16 (a)

when 1 ≤ x ≤ 4.�� < x ≤ �4,y ≤ 0

�x � 1��x2 � 16� ≤ 0

x2�x � 1� � 16�x � 1� ≤ 0

x3 � x2 � 16x � 16 ≤ 0

y ≤ 0 (b)

5 ≤ x < �.y ≥ 36 when x � �2,

�x � 2��x � 5��x � 2� ≥ 0

x3 � x2 � 16x � 20 ≥ 0

x3 � x2 � 16x � 16 ≥ 36

y ≥ 36

37.

Critical numbers:

Test intervals:

Test: Is

By testing an x-value in each test interval in the inequality,we see that the solution set is:

−1−2 210

x

���, �1� � �0, 1�

1 � x2

x> 0?

���, �1�, ��1, 0�, �0, 1�, �1, ��

x � 0, x � ±1

1 � x2

x> 0

1

x� x > 0 38.

Critical numbers:

Test intervals:

Solution interval:

10

14

−1

x

���, 0� � �1

4, ��

�1

4, �� ⇒ 1 � 4x

x< 0

�0, 1

4� ⇒ 1 � 4x

x> 0

���, 0� ⇒ 1 � 4x

x< 0

x � 0, x �1

4

1 � 4x

x< 0

1

x� 4 < 0

39.

Critical numbers:

Test intervals:

Test: Is

By testing an x-value in each test interval in the inequality,we see that the solution set is:

−2 −1 10 2 3 4 5

x

���, �1� � �4, ��

4 � x

x � 1< 0?

���, �1�, ��1, 4�, �4, ��

x � �1, x � 4

4 � x

x � 1< 0

x � 6 � 2�x � 1�x � 1

< 0

x � 6

x � 1� 2 < 0 40.

Critical numbers:

Test intervals:

Solution interval:

−1−2 0 1 2 3

x

��2, 3�

�3, �� ⇒ 6 � 2x

x � 2< 0

��2, 3� ⇒ 6 � 2x

x � 2 > 0

���, �2� ⇒ 6 � 2x

x � 2< 0

x � �2, x � 3

6 � 2x

x � 2≥ 0

x � 12 � 3�x � 2�

x � 2≥ 0

x � 12

x � 2� 3 ≥ 0

Page 99: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.8 Other Types of Inequalities 165

41.

Critical numbers:

Test intervals:

Test: Is

By testing an x-value in each test interval in the inequality,we see that the solution set is:

3 6 9

5

12 15 18

x

�5, 15�

15 � x

x � 5> 0?

���, 5�, �5, 15�, �15, ��

x � 5, x � 15

15 � x

x � 5> 0

3x � 5 � 4�x � 5�x � 5

> 0

3x � 5

x � 5� 4 > 0

3x � 5

x � 5> 4 42.

Critical numbers:

Test intervals:

Solution intervals:

−2 −1

0 21

12

x

���, �1

2� � �1, ��

�1, �� ⇒ 1 � x

1 � 2x< 0

�� 1

2, 1� ⇒ 1 � x

1 � 2x> 0

���, �1

2� ⇒ 1 � x

1 � 2x< 0

x � �1

2, x � 1

1 � x

1 � 2x< 0

5 � 7x � 4�1 � 2x�

1 � 2x< 0

5 � 7x

1 � 2x< 4

43.

Critical numbers:

Test intervals:

Test: Is

By testing an x-value in each test interval in the inequality,we see that the solution set is:

2

−1−2−3−4−5

0

3

x

��5, �32� � ��1, ��

7�x � 1��x � 5��2x � 3�

> 0?

��3

2, �1�, ��1, ��

���, �5�, ��5, �3

2�,

x � �1, x � �5, x � �3

2

7x � 7

�x � 5��2x � 3�> 0

4�2x � 3� � �x � 5��x � 5��2x � 3�

> 0

4

x � 5�

1

2x � 3> 0

4

x � 5>

1

2x � 344.

Critical numbers:

Test intervals:

Solution intervals:

−14

−5

−2

−10−15 0 5

6

10

x

��14, �2� � �6, ��

�6, �� ⇒ 2x � 28

�x � 6��x � 2�> 0

��2, 6� ⇒ 2x � 28

�x � 6��x � 2�< 0

��14, �2� ⇒ 2x � 28

�x � 6��x � 2�> 0

���, �14� ⇒ 2x � 28

�x � 6��x � 2�< 0

x � �14, x � �2, x � 6

2x � 28

�x � 6��x � 2�> 0

5�x � 2� � 3�x � 6�

�x � 6��x � 2�> 0

5

x � 6>

3

x � 2

Page 100: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

166 Chapter 1 Equations, Inequalities, and Mathematical Modeling

45.

Critical numbers:

Test intervals:

Test: Is

By testing an x-value in each test interval in the inequality,we see that the solution set is:

−2−4

−33

4 6 820

4x

��34, 3� � �6, ��

30 � 5x

�x � 3��4x � 3�≤ 0?

���, �3

4�, ��3

4, 3�, �3, 6�, �6, ��

x � 3, x � �3

4, x � 6

30 � 5x

�x � 3��4x � 3�≤ 0

4x � 3 � 9�x � 3��x � 3��4x � 3�

≤ 0

1

x � 3�

9

4x � 3≤ 0

1

x � 3≤

9

4x � 346.

Critical numbers:

Test intervals:

Solution intervals:

−1−2−3−4 10

x

���, �3� � �0, ��

�0, �� ⇒ 3

x�x � 3�> 0

��3, 0� ⇒ 3

x�x � 3�< 0

���, �3� ⇒ 3

x�x � 3�> 0

x � �3, x � 0

3

x�x � 3�≥ 0

1�x � 3� � 1�x�

x�x � 3�≥ 0

1

x≥

1

x � 3

47.

Critical numbers:

Test intervals:

Test: Is

By testing an x-value in each test interval in the inequality,we see that the solution set is:

−1−2−3 0 1 2 3

x

��3, �2� � �0, 3�

x�x � 2��x � 3��x � 3�

≤ 0?

���, �3�, ��3, �2�, ��2, 0�, �0, 3�, �3, ��

x � 0, x � �2, x � ±3

x�x � 2��x � 3��x � 3�

≤ 0

x2 � 2x

x2 � 9≤ 0 48.

Critical numbers:

Test intervals:

Solution intervals:

−1−2−3 0 1 2 3

x

��3, 0� � �2, ��

�2, �� ⇒ �x � 3��x � 2�x

> 0

�0, 2� ⇒ �x � 3��x � 2�x

< 0

��3, 0� ⇒ �x � 3��x � 2�x

> 0

���, �3� ⇒ �x � 3��x � 2�x

< 0

x � �3, x � 0, x � 2

�x � 3��x � 2�

x≥ 0

x2 � x � 6

x≥ 0

Page 101: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.8 Other Types of Inequalities 167

49.

Critical numbers:

Test intervals:

Test: Is

By testing an x-value in each test interval in the inequality,we see that the solution set is:

x

−1

43210

23

���, �1� � ��23, 1� � �3, ��

��3x � 2��x � 3��x � 1��x � 1� < 0?

���, �1�, ��1, �23�, ��2

3, 1�, �1, 3�, �3, ��

x � �23

, x � 3, x � ±1

��3x � 2��x � 3�

�x � 1��x � 1� < 0

�3x2 � 7x � 6�x � 1��x � 1� < 0

5x � 5 � 2x2 � 2x � x2 � 1

�x � 1��x � 1� < 0

5�x � 1� � 2x�x � 1� � �x � 1��x � 1�

�x � 1��x � 1� < 0

5

x � 1�

2xx � 1

� 1 < 0

5

x � 1�

2xx � 1

< 1 50.

Critical numbers:

Test intervals:

Solution intervals:

0 2 64

x

−2−4

1

���, �4�, ��2, 1�, �6, ��

�6, �� ⇒ ��x � 6��x � 2��x � 1��x � 4� < 0

�1, 6� ⇒ ��x � 6��x � 2��x � 1��x � 4� > 0

��2, 1� ⇒ ��x � 6��x � 2��x � 1��x � 4� < 0

��4, �2� ⇒ ��x � 6��x � 2��x � 1��x � 4� > 0

���, �4� ⇒ ��x � 6��x � 2��x � 1��x � 4� < 0

x � �4, x � �2, x � 1, x � 6

��x � 6��x � 2��x � 1��x � 4� ≤ 0

�x2 � 4x � 12�x � 1��x � 4� ≤ 0

3x�x � 4� � x�x � 1� � 3�x � 4��x � 1�

�x � 1��x � 4� ≤ 0

3x

x � 1≤

xx � 4

� 3

51.

−6 12

8

−4

y �3x

x � 2

(a)

(b) y ≥ 6 when 2 < x ≤ 4.

y ≤ 0 when 0 ≤ x < 2.

52.

−15 15

−6

14

y �2�x � 2�

x � 1(a)

y ≤ 0 when �1 < x ≤ 2.

2�x � 2�x � 1

≤ 0

y ≤ 0 (b)

y ≥ 8 when �2 ≤ x < �1.

�6�x � 2�

x � 1≥ 0

�6x � 12

x � 1≥ 0

2�x � 2� � 8�x � 1�

x � 1≥ 0

2�x � 2�

x � 1≥ 8

y ≥ 8

Page 102: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

168 Chapter 1 Equations, Inequalities, and Mathematical Modeling

53.

−6 6

−2

6

y �2x2

x2 � 4(a) when or

This can also be expressed as

(b) for all real numbers

This can also be expressed as �� < x < �.

x.y ≤ 2

x ≥ 2.

x ≥ 2.x ≤ �2y ≥ 1

54.

(a)

y ≥ 1 when 1 ≤ x ≤ 4.

��x � 4��x � 1�

x2 � 4≥ 0

5x � �x2 � 4�

x2 � 4≥ 0

5x

x2 � 4≥ 1

y ≥ 1

y �5x

x2 � 4

(b)

y ≤ 0 when �� < x ≤ 0.

5x

x2 � 4≤ 0 −6 6

−4

4 y ≤ 0

55.

Critical numbers:

Test intervals:

Test: Is

By testing an x-value in each test interval in the inequality,we see that the domain set is: ��2, 2�

4 � x2 ≥ 0?

���, �2�, ��2, 2�, �2, ��

x � ±2

�2 � x��2 � x� ≥ 0

4 � x2 ≥ 0 56.

Critical numbers:

Test intervals:

Domain: ���, �2� � �2, ��

�2, �� ⇒ �x � 2��x � 2� > 0

��2, 2� ⇒ �x � 2��x � 2� < 0

���, �2� ⇒ �x � 2��x � 2� > 0

x � �2, x � 2

�x � 2��x � 2� ≥ 0

x2 � 4 ≥ 0

57.

Critical numbers:

Test intervals:

Test: Is

By testing an x-value in each test interval in the inequality,we see that the domain set is: ���, 3� � �4, ��

�x � 3��x � 4� ≥ 0?

���, 3�, �3, 4�, �4, ��

x � 3, x � 4

�x � 3��x � 4� ≥ 0

x2 � 7x � 12 ≥ 0 58.

Critical numbers:

Test intervals:

Domain: ��4, 4�

�4, �� ⇒ 9�4 � x��4 � x� < 0

��4, 4� ⇒ 9�4 � x��4 � x� > 0

���, �4� ⇒ 9�4 � x��4 � x� < 0

x � �4, x � 4

9�4 � x��4 � x� ≥ 0

144 � 9x2 ≥ 0

59.

Critical numbers:

Test intervals:

Test: Is

By testing an x-value in each test interval in the inequality, we see that the domain set is: ��5, 0� � �7, ��

x�x � 5��x � 7� ≥ 0?

���, �5�, ��5, 0�, �0, 7�, �7, ��

x � 0, x � �5, x � 7

x

�x � 5��x � 7� ≥ 0

x

x2 � 2x � 35≥ 0

Page 103: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.8 Other Types of Inequalities 169

60.

Critical numbers:

Test intervals:

Domain: ��3, 0� � �3, ��

�3, �� ⇒ x

�x � 3��x � 3�> 0

�0, 3� ⇒ x

�x � 3��x � 3�< 0

��3, 0� ⇒ x

�x � 3��x � 3�> 0

���, �3� ⇒ x

�x � 3��x � 3�< 0

x � �3, x � 0, x � 3

x

�x � 3��x � 3�≥ 0

x

x2 � 9≥ 0 61.

Critical numbers:

Test intervals:

By testing an x-value in each test interval in the inequality,we see that the solution set is: ��3.51, 3.51�

���, �3.51�, ��3.51, 3.51�, �3.51, ��

x � ±3.51

0.4�x2 � 12.35� < 0

0.4x2 � 4.94 < 0

0.4x2 � 5.26 < 10.2

62.

Critical numbers:

Test intervals:

Solution set: ��1.13, 1.13�

���, �1.13�, ��1.13, 1.13�, �1.13, ��

±1.13

�1.3x2 � 1.66 > 0

�1.3x2 � 3.78 > 2.12 63.

The zeros are

Critical numbers: ,

Test intervals:

By testing an x-value in each test interval in the inequality,we see that the solution set is: ��0.13, 25.13�

�25.13, ��

���, �0.13�, ��0.13, 25.13�,

x � 25.13 x � �0.13

x ��12.5 ± ��12.5�2 � 4��0.5��1.6�

2��0.5�.

�0.5x2 � 12.5x � 1.6 > 0

64.

Critical numbers:

Test intervals:

Solution set: ��4.42, 0.42�

���, �4.42�, ��4.42, 0.42�, �0.42, ��

�4.42, 0.42

1.2x2 � 4.8x � 2.2 < 0

1.2x2 � 4.8x � 3.1 < 5.3 65.

Critical numbers:

Test intervals:

By testing an x-value in each test interval in the inequality,we see that the solution set is: �2.26, 2.39�

���, 2.26�, �2.26, 2.39�, �2.39, ��

x � 2.39, x � 2.26

�7.82x � 18.68

2.3x � 5.2> 0

1 � 3.4�2.3x � 5.2�

2.3x � 5.2> 0

1

2.3x � 5.2� 3.4 > 0

1

2.3x � 5.2> 3.4

66.

23.46 � 17.98x

3.1x � 3.7> 0

2 � 5.8�3.1x � 3.7�

3.1x � 3.7> 0

2

3.1x � 3.7> 5.8

Critical numbers:

Test intervals:

Solution interval: �1.19, 1.30�

�1.30, �� ⇒ 23.46 � 17.98x

3.1x � 3.7< 0

�1.19, 1.30� ⇒ 23.46 � 17.98x

3.1x � 3.7> 0

���, 1.19� ⇒ 23.46 � 17.98x

3.1x � 3.7< 0

x � 1.19, x � 1.30

Page 104: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

170 Chapter 1 Equations, Inequalities, and Mathematical Modeling

67.

(a)

It will be back on the ground in 10 seconds.

(b)

4 < t < 6 seconds

�t � 4��t � 6� < 0

t2 � 10t � 24 < 0

�16�t2 � 10t � 24� > 0

�16t2 � 160t � 384 > 0

�16t2 � 160t > 384

t � 0, t � 10

�16t�t � 10� � 0

�16t2 � 160t � 0

s � �16t2 � v0t � s0 � �16t2 � 160t 68.

(a)

It will be back on the ground in 8 seconds.

(b)

Critical numbers:

Test intervals:

Solution set:and 4 � 2�2 seconds < t ≤ 8 seconds0 seconds ≤ t < 4 � 2�2 seconds

�4 � 2�2, �����, 4 � 2�2�, �4 � 2�2, 4 � 2�2�,

4 � 2�2, 4 � 2�2

�16t2 � 128t � 128 < 0

�16t2 � 128t < 128

t � 8 � 0 ⇒ t � 8

�16t � 0 ⇒ t � 0

�16t�t � 8� � 0

�16t2 � 128t � 0

s � �16t2 � v0t � s0 � �16t 2 � 128t

69.

By the Quadratic Formula we have:

Critical numbers:

Test: Is

Solution set:

13.8 meters ≤ L ≤ 36.2 meters

25 � 5�5 ≤ L ≤ 25 � 5�5

�L2 � 50L � 500 ≥ 0?

L � 25 ± 5�5

�L2 � 50L � 500 ≥ 0

L�50 � L� ≥ 500

LW ≥ 500

2L � 2W � 100 ⇒ W � 50 � L 70.

By the Quadratic Formula we have:

Critical numbers:

Test:

Solution set:

45.97 feet ≤ L ≤ 174.03 feet

110 � 10�41 ≤ L ≤ 110 � 10�41

Is �L2 � 220L � 8000 ≥ 0?

L � 110 ± 10�41

�L2 � 220L � 8000 ≥ 0

L�220 � L� ≥ 8000

LW ≥ 8000

2L � 2W � 440 ⇒ W � 220 � L

71. and

Critical numbers: (These wereobtained by using the Quadratic Formula.)

Test intervals:

By testing values in each test interval in the inequality, wesee that the solution set is or

The price per unit is

For For Therefore, for 40,000 ≤ x ≤ 50,000, $50.00 ≤ p ≤ $55.00.

x � 50,000, p � $50.x � 40,000, p � $55.

p �Rx

� 75 � 0.0005x.

40,000 ≤ x ≤ 50,000.�40,000, 50,000�

x-

�50,000, ���0, 40,000�, �40,000, 50,000�,

x � 40,000, x � 50,000

�0.0005x2 � 45x � 1,000,000 ≥ 0

�0.0005x2 � 45x � 250,000 ≥ 750,000

P ≥ 750,000

� �0.0005x2 � 45x � 250,000

� �75x � 0.0005x2� � �30x � 250,000�

P � R � C

C � 30x � 250,000R � x�75 � 0.0005x� 72. What is the price per unit?

When

When

Solution interval: $30.00 ≤ p ≤ $32.00

R � $3,000,000 ⇒ 3,000,000100,000

� $30 per unit

x � 100,000:

R � $2,880,000 ⇒ 2,880,00090,000

� $32 per unit

x � 90,000:

Page 105: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Section 1.8 Other Types of Inequalities 171

73.

(a)

(b) will be greater than 75% whenwhich corresponds to 2011.

(c) when t � 30.41.C � 75

t � 31,C

0 230

80

C � 0.0031t3 � 0.216t 2 � 5.54t � 19.1, 0 ≤ t ≤ 23

(d) will be between 85% and 100%when is between 37 and 42. Thesevalues correspond to the years 2017 to2022.

(e) when or

(f ) The model is a third-degree polynomial and ast →�, C →�.

37 ≤ t ≤ 42.36.82 ≤ t ≤ 41.8985 ≤ C ≤ 100

tC

24 70.5

26 71.6

28 72.9

30 74.6

32 76.8

34 79.6

Ct

36 83.2

37 85.4

38 87.8

39 90.5

40 93.5

41 96.8

42 100.4

43 104.4

Ct

74. (a)

(b)

The minimum depth is 3.83 inches.

3.83 ≤ d

14.67 ≤ d2

2472.1 ≤ 168.5d2

2000 ≤ 168.5d2 � 472.1

Depth of the beam

Max

imum

saf

e lo

ad

d

L

4 6 8 10 12

5,000

10,000

15,000

20,000

25,000

75.

Since we have

Since the only critical number is Theinequality is satisfied when R1 ≥ 2 ohms.

R1 � 2.R1 > 0,

R1 � 2

2 � R1

≥ 0.

2R1

2 � R1

� 1 ≥ 0

2R1

2 � R1

≥ 1

R ≥ 1,

2R1

2 � R1

� R

2R1 � R�2 � R1�

2R1 � 2R � RR1

1

R�

1

R1

�1

2d 4 6 8 10 12

Load 2223.9 5593.9 10,312 16,378 23,792

76. (a)

So the number of master’s degrees earned by womenexceeded 220,000 in 1995.

(c)

So the number of master’s degrees earned by womenwill exceed 320,000 in 2006.

� 320 ⇒ t � 16.2

N � �0.03t 2 � 9.6t � 172

� 220 ⇒ t � 5

N � �0.03t 2 � 9.6t � 172 (b) and (d)

t

N

Mas

ter's

deg

rees

ear

ned

(in

thou

sand

s)

Year (0 ↔ 1990)

2 6 10 14 18

160

200

240

280

320

N = 220

N = 320

Page 106: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

172 Chapter 1 Equations, Inequalities, and Mathematical Modeling

77. True

The test intervals are and�4, ��.

���, �3�, ��3, 1�, �1, 4�,

x3 � 2x2 � 11x � 12 � �x � 3��x � 1��x � 4�

78. True

The y-values are greater than zero for all values of x.

79.

To have at least one real solution, Thisoccurs when This can be written as���, �4� � �4, ��.

b ≤ �4 or b ≥ 4.b2 � 16 ≥ 0.

x2 � bx � 4 � 0 80.

To have at least one real solution,

This inequality is true for all real values of b. Thus, theinterval for b such that the equation has at least one realsolution is ���, ��.

b2 � 16 ≥ 0.

b2 � 4�1���4� ≥ 0

x2 � bx � 4 � 0

81.

To have at least one real solution,

Critical numbers:

Test intervals:

Test: Is

Solution set: ���, �2�30� � �2�30, ��b2 � 120 ≥ 0?

���, �2�30 �, ��2�30, 2�30 �, �2�30, ��

b � ±�120 � ±2�30

�b � �120 ��b � �120 � ≥ 0

b2 � 120 ≥ 0

b2 � 4�3��10� ≥ 0.

3x2 � bx � 10 � 0 82.

To have at least one real solution,

This occurs when or Thus,the interval for b such that the equation has at least one real solution is ���, �2�10� � �2�10, ��.

b ≥ 2�10.b ≤ �2�10

b2 � 40 ≥ 0.

b2 � 4�2��5� ≥ 0

2x2 � bx � 5 � 0

83. (a) If a > 0 and c ≤ 0, then b can be any real number. Ifand then for to be greater than

or equal to zero, b is restricted to or

(b) The center of the interval for b in Exercises 79–82 is 0.

b > 2�ac.b < �2�ac

b2 � 4acc > 0,a > 084. (a)

(b)

(c) The real zeros of the polynomial

ax

b

+ − +

− − +

− + +

x � a, x � b

85. 4x2 � 20x � 25 � �2x � 5�2 86.

� �x � 7��x � 1�

�x � 3�2 � 16 � ��x � 3� � 4���x � 3� � 4�

87.

� �x � 2��x � 2��x � 3�

x2�x � 3) � 4�x � 3� � �x2 � 4��x � 3� 88.

� 2x�x � 3��x2 � 3x � 9�

2x4 � 54x � 2x�x3 � 27�

89.

� 2x2 � x

� �2x � 1��x�

Area � �length��width� 90.

� 32b2 � b

� 12�b��3b � 2�

Area �12�base��height�

Page 107: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Review Exercises for Chapter 1 173

1.

–3 –2 –1 1 2 3

–5

–4

–3

–2

–1

1

x

y

y � 3x � 5 2.

x−2

−2−4−6 2 4

4

6

8

10

y

y � �12x � 2

x 0 1 2

y 1�2�5�8�11

�1�2 x �4 �2 0 2 4

y 4 3 2 1 0

3.

–3 –2 –1 1 2 54

–3

–2

4

5

x

y

y � x2 � 3x 4.

x

−3

5431

1

−2

−4

−9

−3−4−5 −1

y

y � 2x2 � x � 9

x 0 1 2 3 4

y 4 0 0 4�2�2

�1 x �2 �1 0 1 2 3

y 1 �6 �9 �8 �3 6

5.

Line with x-intercept andy-intercept

–5 –4 –3 –1 1 2 3

–2

1

3

4

5

6

x

y

�0, 3���3

2, 0�

y � 2x � 3

y � 2x � 3 � 0 6.

Line with x-intercept andy-intercept

–5 –4 –3 1 2 3

–6

–5

–4

–3

1

2

x

y

�0, �3���2, 0�

y � �32x � 3

2y � �3x � 6

3x � 2y � 6 � 0 7.

Domain:

–2 –1 1 2 3 4 5 6

–2

1

3

4

5

6

x

y

���, 5�

y � �5 � x

x 5 4 1

y 0 1 2 3

�4x 0 1

y 1 3 5�1

�1�2

x 0 1

y 0 �92�3�

32

32

�1�2�3

Review Exercises for Chapter 1

Page 108: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

174 Chapter 1 Equations, Inequalities, and Mathematical Modeling

9.

is a parabola.

–3 –2 –1 1 2 3

–5

–4

–3

–2

1

x

y

y � �2x2

y � 2x2 � 0 10. is a parabola.

–2 –1 1 2 3 5 6

–4

–3

–2

x

y

y � x2 � 4x

x 0

y 0 �8�2

±2±1

x �1 0 1 2 3 4

y 5 0 �3 �4 �3 0

11.

x-intercepts:

y-intercept:

The intercepts are and The intercept is �0, 5�.y-�5, 0�.�1, 0�x-

y � �0 � 3�2 � 4 ⇒ y � 9 � 4 ⇒ y � 5

⇒ x � 5 or x � 1

0 � �x � 3�2 � 4 ⇒ �x � 3�2 � 4 ⇒ x � 3 � ±2 ⇒ x � 3 ± 2

y � �x � 3�2 � 4

12.

For

For

or

The x-intercepts are and the y-intercept is �0, �2�.��4, 0�;�2, 0�

y � �2y � �0 � 1� � 3

y � �x � 1� � 3

x � 1 < 0, 0 � ��x � 1� � 3, or �4 � x.

x � 1 > 0, 0 � x � 1 � 3, or 2 � x.

0 � �x � 1� � 3

y � �x � 1� � 3

13.

Intercepts:

No y-axis symmetry

No x-axis symmetry

No origin symmetry�y � �4��x� � 1 ⇒ y � �4x � 1 ⇒

�y � �4x � 1 ⇒ y � 4x � 1 ⇒

y � �4��x� � 1 ⇒ y � 4x � 1 ⇒

�14, 0�, �0, 1�

x1−1−2−3−4

4

2−1

3

1

4

−2

−3

−4

yy � �4x � 1

14.

Intercepts:

No symmetry

�65, 0�, �0, �6�

x6

1

2 3 4 5−1−2−1

−2

−3

−4

−5

−6

−7

yy � 5x � 6

8.

−3 −2 −1 1 2 3 4 5

2

−2

3

4

5

6

x

y

y � �x � 2, domain: ��2, ��

x �2 0 2 7

y 0 2 3�2

Page 109: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Review Exercises for Chapter 1 175

15.

Intercepts:

y-axis symmetry

No x-axis symmetry

No origin symmetry

x1

6

−13 42−1−3−4

3

−2

1

2

4

y

�y � 5 � ��x�2 ⇒ y � �5 � x2 ⇒

�y � 5 � x2 ⇒ y � �5 � x2 ⇒

y � 5 � ��x�2 ⇒ y � 5 � x2 ⇒

�±�5, 0�, �0, 5�

y � 5 � x2 16.

Intercepts:

y-axis symmetry

x

2

42−4−6 6 8

−6

−12

−4

−2

y

�±�10, 0�, �0, �10�

y � x2 � 10

17.

Intercepts:

No y-axis symmetry

No x-axis symmetry

No origin symmetry

x1 2 4

7

−13−2−3−4

1

2

4

5

6

y

�y � ��x�3 � 3 ⇒ y � x3 � 3 ⇒

�y � x3 � 3 ⇒ y � �x3 � 3 ⇒

y � ��x�3 � 3 ⇒ y � �x3 � 3 ⇒

�� 3�3, 0�, �0, 3�

y � x3 � 3 18.

Intercepts:

No symmetry

x

2

42 6−4−6

−8

−10

y

� 3��6, 0�, �0, �6�

y � �6 � x3

19.

Domain:

Intercepts:

No y-axis symmetry

No x-axis symmetry

No origin symmetry�y � ��x � 5 ⇒ y � ���x � 5 ⇒

�y � �x � 5 ⇒ y � ��x � 5 ⇒

y � ��x � 5 ⇒

��5, 0�, �0, �5���5, ��

x1

7

−6−1

−−5

1

3

4

5

6

−1−24 2−3

yy � �x � 5

20.

Intercepts:

y-axis symmetry

�0, 9�

y

x−3−6−9 3 6 9

−3

3

6

9

12

15

y � �x� � 9 21.

Center:

Radius: 3

�0, 0�

–4 –2 –1 1 2 4

–4

–2

–1

1

2

4

x(0, 0)

yx2 � y2 � 9

Page 110: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

176 Chapter 1 Equations, Inequalities, and Mathematical Modeling

22.

Center:

Radius: 2

�0, 0�

x

3

3−3

1

1−1−1

−3

(0, 0)

yx2 � y2 � 4 23.

Center:

Radius: 4

��2, 0�

�x � ��2��2 � �y � 0�2 � 42

–8 –4 –2 4

–6

–2

2

6

x(−2, 0)

y�x � 2�2 � y2 � 16

24.

Center:

Radius: 9

�0, 8�

x4

8

10

4

−2−4−6−8

2

6

(0, 8)

6 8

10

12

14

16

18

yx2 � �y � 8�2 � 81 25.

Center:

Radius: 6

�12, �1�

�x �12�2

� �y � ��1��2 � 62

x2

8

8−2

−8

2

4

−4

−2−4−6 4

( (12, −1

y�x �12�2

� �y � 1�2 � 36

26.

Center:

Radius: 10

��4, 32� �x � ��4��2 � � y �

32�2

� 100

x9

3

3−3−6−9−3

−15

6

9

15

( (32

−4,

−6

y �x � 4�2 � � y �32�2

� 100 27. Endpoints of a diameter: and

Center:

Radius:

Standard form:

�x � 2�2 � �y � 3�2 � 13

�x � 2�2 � �y � ��3��2 � ��13�2

r � ��2 � 0�2 � ��3 � 0�2 � �4 � 9 � �13

�0 � 42

, 0 � ��6�

2 � � �2, �3�

�4, �6��0, 0�

28. Endpoints of a diameter:

Center:

Radius:

Standard form:

�x � 1�2 � �y �132 �

2

�854

�x � 1�2 � �y � ��132 ��

2

� ��854 �

2

r ���1 � ��2��2 � ��132

� ��3��2

��9 �494

��854

��2 � 42

, �3 � ��10�

2 � � �1, �132 �

��2, �3� and �4, �10�

29.

(a)

(c) When pounds.F �504 � 12.5x � 10,

0 ≤ x ≤ 20F �54x,

(b)

x

F

4 8 12 2416 20

5

10

15

20

25

Length (in inches)

Forc

e (i

n po

unds

)

30x 0 4 8 12 16 20

F 0 5 10 15 20 25

Page 111: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Review Exercises for Chapter 1 177

30. (a)

(b) The number of stores was 1300 in 2003.z � 9.94;

t

1800

Num

ber

of T

arge

t sto

res

Year (4 ↔ 1994)

4 6 8 10 12

800

1000

1200

1400

1600

N 31.

All real numbers are solutions.

0 � 0 Identity

2 � 4x � x2 � 2 � 4x � x2

6 � �x2 � 4x � 4� � 2 � 4x � x2

6 � �x � 2�2 � 2 � 4x � x2

32.

Conditional equation

x � 4

3x � 12

3x � 6 � 2x � 2x � 6

3�x � 2� � 2x � 2�x � 3� 33.

All real numbers are solutions.

0 � 0 Identity

�x3 � x2 � 7x � 3 � �x3 � x2 � 7x � 3

�x3 � 7x � x2 � 3 � �x3 � x2 � 7x � 7 � 4

�x3 � x�7 � x� � 3 � x��x2 � x� � 7�x � 1� � 4

34.

Conditional equation

3x2 � x � 19 � 0

6x2 � 2x � 38 � 0

3x2 � 12x � 24 � �3x2 � 10x � 14

3x2 � 12x � 24 � �10x � 20 � 3x2 � 6

3�x2 � 4x � 8� � �10�x � 2� � 3x2 � 6 35.

x � 20

3x � 2x � 10 � 10

3x � 2�x � 5� � 10

36.

x � �92

2x � �9

4x � 14 � 2x � 5

4x � 2�7 � x� � 5 37.

x � �12

10x � �5

4x � 9 � �6x � 4

4x � 12 � 3 � 8 � 6x � 4

4�x � 3� � 3 � 2�4 � 3x� � 4 38.

x � �173

�32x �

172

12x �32 � 2x � 2 � 5

12�x � 3� � 2�x � 1� � 5

39.

x � �5

�4x � 20

x � 15 � 5x � 5

5�x5

� 3� � �x � 1�5

x5

� 3 �2x2

� 1 40.

x � 18

�x � �18

8x � 6 � 3x � 12x � 24

2�4x � 3� � 3x � 12x � 24

4x � 3

6�

x4

� x � 2 41.

x � 9

8x � 72

18x � 72 � 10x

18�x � 4� � 10x

18x

�10

x � 4

42.

x �113

�3x � �11

10x � 15 � 13x � 26

5�2x � 3� � 13�x � 2�

5

x � 2�

132x � 3

43.

x-intercept:

y-intercept:

The x-intercept is and the y-intercept is �0, �1�.�13, 0�

y � 3�0� � 1 ⇒ y � �1

0 � 3x � 1 ⇒ x �13

y � 3x � 1

Page 112: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

178 Chapter 1 Equations, Inequalities, and Mathematical Modeling

44.

The x-intercept is and the y-intercept is �0, 6�.�65, 0�

65 � x

y � 6 �6 � �5x

y � �5�0� � 6 0 � �5x � 6

y � �5x � 6 y � �5x � 6 45.

x-intercept:

y-intercept:

The x-intercept is and the y-intercept is �0, �8�.�4, 0�

y � 2�0 � 4� ⇒ y � �8

0 � 2�x � 4� ⇒ x � 4

y � 2�x � 4�

46.

The x-intercept is and the y-intercept is �0, 4�.��17, 0�

�17

� x

�4 � 28x

y � 4 0 � 28x � 4

y � 4�7�0� � 1� 0 � 4�7x � 1�

y � 4�7x � 1� y � 4�7x � 1� 47.

x-intercept:

y-intercept:

The x-intercept is and the y-intercept is �0, 23�.�43, 0�

y � �12

�0� �23

⇒ y �23

0 � �12

x �23

⇒ x �2312

�43

y � �12

x �23

48.

The x-intercept is and the y-intercept is �0, �14�.�1

3, 0�

13

� x

y � �14

43

�14

�43

�34

x

y �34

�0� �14

0 �34

x �14

y �34

x �14

y �34

x �14

49.

x-intercept:

y-intercept:

The x-intercept is and the y-intercept is �0, � 519�.�2, 0�

3.8y � 0.5�0� � 1 � 0 ⇒ y ��13.8

� �5

19

3.8�0� � 0.5x � 1 � 0 ⇒ x �1

0.5� 2

3.8y � 0.5x � 1 � 0

50.

The x-intercept is and the y-intercept is �0, 0.8�.�0.6, 0�

y � 0.8 x � 0.6

1.5y � 1.2 2x � 1.2

1.5y � 2�0� � 1.2 � 0 1.5�0� � 2x � 1.2 � 0

1.5y � 2x � 1.2 � 0 1.5y � 2x � 1.2 � 0 51.

The height is 10 inches.

10 � h

188.40 � 18.84h

244.92 � 56.52 � 18.84h

244.92 � 2�3.14��3�2 � 2�3.14��3�h

52.

For F �95�100 �

1609 � � 212�FC � 100�:

F �95�C �

1609 �

59F � C �160

9

C �59F �

1609

53. Verbal Model:

Labels: Let September’s profit. Then

Equation:

Answer: September profit: $325,000, October profit: $364,000

x � 0.12x � 364,000

x � 325,000

2.12x � 689,000

x � �x � 0.12x� � 689,000

x � 0.12x � October’s profit.x �

September’s profit � October’s profit � 689,000

Page 113: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Review Exercises for Chapter 1 179

54. Model:

Labels: Original price

Discount rate

Sale price

Equation:

The percent discount is 20%.

x � 0.2

1 � x � 0.8

425�1 � x� � 340

� 340

� x

� 340 � 85 � 425

�Original price��1 � discount rate� � �sale price� 55. Let height of the streetlight.

By similar triangles we have:

The streetlight is 24 feet tall.

x � 24

5x � 120

x

20�

65

x

15 ft5 ft

6 ft

x �

56. Let the number of members in the group.

Cost per member

If two more members join the group, the cost per member will be

Since this new cost is $4000 less than the original cost:

There are four members presently in the group.

x � 4 � 0 ⇒ x � 4

x � 6 � 0 ⇒ x � �6, extraneous

0 � �x � 6��x � 4�

0 � x2 � 2x � 24

12x � 24 � x2 � 2x � 12x

12�x � 2� � x�x � 2� � 12x Divide both sides by 4000.

48,000�x � 2� � 4000x�x � 2� � 48,000x

48,000

x� 4000 �

48,000

x � 2

48,000

x � 2.

�48,000

x

x �

57. Let the number of original investors.

Each person’s share is If three more people invest, each person’s share is

Since this is $2500 less than the original cost, we have:

extraneous or

There are currently nine investors.

x � 9x � �12,

�2500�x � 12��x � 9� � 0

�2500�x2 � 3x � 108� � 0

�2500x2 � 7500x � 270,000 � 0

90,000x � 270,000 � 2500x2 � 7500x � 90,000x

90,000�x � 3� � 2500x�x � 3� � 90,000x

90,000

x� 2500 �

90,000x � 3

90,000x � 3

.90,000

x.

x �

Page 114: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

180 Chapter 1 Equations, Inequalities, and Mathematical Modeling

58.

Using the positive value for r, we have miles per hour. The average speed on the trip home was miles per hour.r � 8 � 56

r � 48

0 � �r � 48��r � 56�

0 � r2 � 8r � 2688

336r � 2688 � 336r � r2 � 8r

6�r � 8��56� � 6r�56� � r�r � 8�

56

r�

56

r � 8�

1

6 Convert minutes to portion of an hour.

Time to work � time from work � 10 minutes

Time �distance

rate

Rate Time Distance

To work r 56

From work 5656

r � 8r � 8

56r

59. Let the number of liters of pure antifreeze.

x �2

0.70�

20

7� 2

6

7 liters

0.70x � 2

3 � 0.30x � 1.00x � 5

0.30�10 � x� � 1.00x � 0.50�10�

30% of �10 � x� � 100% of x � 50% of 10

x �

60. Model:

Labels: Amount invested at amount invested at

Interest from interest from total interest $305

Equation:

The amount invested at was $2500 and the amount invested at was 6000 � 2500 � $3500.512%41

2%

x � 2500

�0.01x � �25

0.045x � 330 � 0.055x � 305

0.045x � 0.055�6000 � x� � 305

�512% � �6000 � x��0.055��1�,41

2% � x�0.045��1�,

512% � 6000 � x41

2% � x,

�Interest from 412%� � �interest from 51

2%� � �total interest�

61.

3V

�r 2� h

3V � �r 2 h

V �1

3�r 2h 62.

m �2Ev2

mv2 � 2E

E �12

mv2

Page 115: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Review Exercises for Chapter 1 181

63.

or 40 minutes t �23 hour

15t � 10

55t � 40t � 10

64.

h �81�

9�� 9 feet

81� � � �3�2h

V � �r2hrate time distance

1st car 40 mph t 40t

2nd car 55 mph t 55t

65.

x � 3 � 0 ⇒ x � 3

2x � 5 � 0 ⇒ x � �52

0 � �2x � 5��x � 3�

0 � 2x2 � x � 15

15 � x � 2x2 � 0 66.

x � 4 � 0 ⇒ x � 4

2x � 7 � 0 ⇒ x � �72

�2x � 7��x � 4� � 0

2x2 � x � 28 � 0 67.

±�2 � x

2 � x2

6 � 3x2

68.

4x � 5 � 0 ⇒ x � �54

4x � 5 � 0 ⇒ x �54

�4x � 5��4x � 5� � 0

16x2 � 25 � 0

16x2 � 25 69.

x � �4 ± 3�2

x � 4 � ±�18

�x � 4�2 � 18 70.

x � 8 ± �15

x � 8 � ±�15

�x � 8�2 � 15

71.

x � 6 ± �6

x � 6 � ±�6

�x � 6�2 � 6

x2 � 12x � 36 � �30 � 36

x2 � 12x � �30

x2 � 12x � 30 � 0 72.

� �3 ± 2�3

��6 ± 4�3

2

��6 ± �48

2

��6 ± �62 � 4�1���3�

2�1�

x ��b ± �b2 � 4ac

2a

x2 � 6x � 3 � 0 73.

��5 ± �241

4

x ��5 ± �52 � 4�2���27�

2�2�

2x2 � 5x � 27 � 0

�2x2 � 5x � 27 � 0

74.

�3 ± �249

6�

12

±�249

6

����3� ± ���3�2 � 4�3���20�

2�3�

x ��b ± �b2 � 4ac

2a

�20 � 3x � 3x2 � 0 75.

(a) when feet and feet.

(b)

(c) The bending moment is greatest when feet.x � 10

00

22

55,000

x � 20x � 0500x�20 � x� � 0

M � 500x�20 � x�

Page 116: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

182 Chapter 1 Equations, Inequalities, and Mathematical Modeling

76. (a)

(b)

(c)

The ball will hit the ground in about two seconds.

2.052 or �0.1767

��30 ± �1271.2

�32

t ��30 ± �302 � 4��16��5.8�

2��16�

�16t 2 � 30t � 5.8 � 0

h�1� � �16 � 12 � 30 � 1 � 5.8 � 19.8 feet

h�t� � �16t 2 � 30t � 5.8

77. 6 � ��4 � 6 � 2i 78. 3 � ��25 � 3 � 5i 79. i2 � 3i � �1 � 3i

80. �5i � i2 � �1 � 5i 81. �7 � 5i� � ��4 � 2i� � �7 � 4� � �5i � 2i� � 3 � 7i

82. � �2��2

2i� � ��2 i��2

2�

�2

2i� � ��2

2�

�2

2i� �

�2

2�

�2

2i �

�2

2�

�2

2i

83. 5i�13 � 8i� � 65i � 40i2 � 40 � 65i 84.

� 17 � 28i

� 5 � 28i � 12

�1 � 6i��5 � 2i� � 5 � 2i � 30i � 12i2

85.

� �4 � 46i

�10 � 8i��2 � 3i� � 20 � 30i � 16i � 24i2 86.

� 9 � 20i

� 20i � 9i2

� i�20 � 9i�

i�6 � i��3 � 2i� � i�18 � 12i � 3i � 2i2�

87.

�2317

�1017

i

�23 � 10i

17

�24 � 10i � i2

16 � 1

6 � i4 � i

�6 � i4 � i

�4 � i4 � i

88.

�1726

�7i26

�17 � 7i

26

�15 � 3i � 10i � 2i2

25 � i2

3 � 2i5 � i

�3 � 2i5 � i

�5 � i5 � i

89.

�2113

�1

13i

� � 813

� 1� � �1213

i � i�

�8

13�

1213

i � 1 � i

�8 � 12i4 � 9

�2 � 2i1 � 1

4

2 � 3i�

21 � i

�4

2 � 3i�

2 � 3i2 � 3i

�2

1 � i�

1 � i1 � i 90.

�9 � 83i

85�

985

�83i85

�18 � 81i � 2i � 9i2

4 � 81i2

��9 � i

�2 � 9i�

��2 � 9i���2 � 9i�

�1 � 4i � 10 � 5i2 � 8i � i � 4i2

1

2 � i�

51 � 4i

��1 � 4i� � 5�2 � i�

�2 � i��1 � 4i�

Page 117: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Review Exercises for Chapter 1 183

91.

� ±�1

3 i � ±

�3

3i

x � ±��1

3

x2 � �1

3

3x2 � �1

3x2 � 1 � 0 92.

x � ±12

i

x2 � �14

8x2 � �2

2 � 8x2 � 0 93.

x � 1 ± 3i

x � 1 � ±��9

�x � 1�2 � �9

x2 � 2x � 1 � �10 � 1

x2 � 2x � 10 � 0

94.

��3 ± 3i�71

12� �

14

±�71

4i

��3 ± ��639

12

��3 ± �32 � 4�6��27�

2�6�

x ��b ± �b2 � 4ac

2a

6x2 � 3x � 27 � 0 95.

x � 0 or 5 � x �12

5

x3 � 0 or 5x � 12 � 0

x3�5x � 12� � 0

5x4 � 12x3 � 0 96.

4x � 6 � 0 ⇒ x �3

2

x2 � 0 ⇒ x � 0

x2�4x � 6� � 0

4x3 � 6x2 � 0

97.

or

x � ±�3 x � ±�2

x2 � 3 x2 � 2

x2 � 3 � 0 x2 � 2 � 0

�x2 � 2��x2 � 3� � 0

x4 � 5x2 � 6 � 0 98.

9x2 � 4 � 0 ⇒ x � ± 23

x � 0

x � 3 � 0 ⇒ x � �3

�x � 3��x��9x2 � 4� � 0

�x � 3��9x3 � 4x� � 0

9x3�x � 3� � 4x�x � 3� � 0

9x 4 � 27x3 � 4x2 � 12x � 0

99.

x � 5

x � 4 � 9

��x � 4 �2� �3�2

�x � 4 � 3 100.

x � 66

x � 2 � 64

�x � 2 � 8

�x � 2 � 8 � 0

101.

extraneous or extraneous

No solution

x � 11,x � 3,

�x � 3��x � 11� � 0

x2 � 14x � 33 � 0

x2 � 2x � 1 � 16�x � 2�

�x � 1�2 � ��4�x � 2 �2

x � 1 � �4�x � 2

2x � 3 � 4 � 4�x � 2 � x � 2

��2x � 3 �2� �2 � �x � 2 �2

�2x � 3 � �x � 2 � 2

Page 118: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

184 Chapter 1 Equations, Inequalities, and Mathematical Modeling

102.

x �38 � 5�3

24, extraneous

x �38 � 5�3

24

�1824 ± �172,800

1152�

1824 ± 240�3

1152 x �

���1824� ± ���1824�2 � 4�576��1369�2�576�

576x2 � 1824x � 1369 � 0

576x2 � 1680x � 1225 � 144�x � 1�

24x � 35 � 12�x � 1

25x � 36 � 12�x � 1 � x � 1

5�x � 6 � �x � 1

5�x � �x � 1 � 6

103.

x � 126 or x � �124

x � 1 ± 125

x � 1 � ±�253

�x � 1�2 � 253

�x � 1�23 � 25

�x � 1�23 � 25 � 0 104.

x � 79

x � 2 � 81

x � 2 � 2743

�x � 2�34 � 27

105.

or

x � �4

5x2 � 20x � 1 � 0 �x � 4�12 � 0

�x � 4�12�5x2 � 20x � 1� � 0

�x � 4�12�1 � 5x�x � 4�� � 0

�x � 4�12 � 5x�x � 4�32 � 0

x � �2 ±�95

5

x ��20 ± 2�95

10

x ��20 ± �400 � 20

10

x ��20 ± �400 � 20

10

106.

3x � 2 � 0 ⇒ x � �2

3

3x � 2 � 0 ⇒ x �2

3

x � 2 � 0 ⇒ x � �2

x � 2 � 0 ⇒ x � 2

�x � 2�13�x � 2�13�3x � 2��3x � 2� � 0

�x2 � 4�13�9x2 � 4� � 0

�x2 � 4�13�8x2 � x2 � 4� � 0

8x2�x2 � 4�13 � �x2 � 4�43 � 0 107.

±�10 � x

10 � x2

5x � 10 � x2 � 2x � 3x

5�x � 2� � 1�x��x � 2� � 3x

5x

� 1 �3

x � 2

Page 119: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Review Exercises for Chapter 1 185

108.

0 � x � 3 ⇒ x � 3

0 � 3x � 10 ⇒ x � �10

3

0 � �3x � 10��x � 3�

0 � 3x2 � x � 30

14x � 30 � 3x2 � 15x

6�x � 5� � 8x�3x�x � 5�

x�x � 5�6

x� x�x � 5�

8

x � 5� 3x�x � 5� 109.

x �43

9x � 12

15x � 5x � 10 � x � 2

3�5x� � 5�x � 2� � x � 2

3

x � 2�

1x

�15x

110.

x � �25

�3x � 75

17x � 25 � 20x � 100

12x � 5x � 25 � 20x � 100

x�x � 5�12

x � 5�

x�x � 5�5x

�x�x � 5�20

x111.

x � �51 or 5 � x � 15

x � 5 � �10 or x � 5 � 10

�x � 5� � 10

112.

x � 2 x � �5

2x � 4 2x � �10

2x � 3 � 7 or 2x � 3 � �7

�2x � 3� � 7 113.

or

The only solutions to the original equation are or �x � �3 and x � �1 are extraneous.�x � 1.

x � 3

x � �3 or x � 1 x � 3 or x � �1

�x � 3��x � 1� � 0 �x � 3��x � 1� � 0

x2 � 2x � 3 � 0 x2 � 2x � 3 � 0

x2 � 3 � �2x x2 � 3 � 2x

�x2 � 3� � 2x

114.

x � 2 � 0 ⇒ x � �2, extraneous

x � 3 � 0 ⇒ x � 3

�x � 3��x � 2� � 0

x2 � x � 6 � 0

x2 � 6 � x

�x2 � 6� � x

x � 3 � 0 ⇒ x � �3, extraneous

x � 2 � 0 ⇒ x � 2

�x � 3��x � 2� � 0

x2 � x � 6 � 0

or ��x2 � 6� � x

115.

143,203 units

x � 143,202.5

0.001x � 143.2025

0.001x � 2 � 145.2025

�0.001x � 2 � 12.05

�12.05 � ��0.001x � 2

29.95 � 42 � �0.001x � 2 116. (a)

(b) There were 800 daily evening papers about 1997.

(c)

t 27.98 ⇒ 1998

t �148.04�23

t 32 ��681�4.6

148.04

�681 � �4.6t 32

800 � 1481 � 4.6t 32

0 300

1500

Page 120: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

186 Chapter 1 Equations, Inequalities, and Mathematical Modeling

117. Interval:

Inequality:

The interval is bounded.

�7 < x ≤ 2

��7, 2� 118. Interval:

Inequality:

The interval is unbounded.

4 < x < �

�4, �� 119. Interval:

Inequality:

The interval is unbounded.

x ≤ �10

���, �10�

120. Interval:

Inequality:

The interval is bounded.

�2 ≤ x ≤ 2

��2, 2� 121.

���, 12�

x ≤ 12

2x ≤ 24

9x � 8 ≤ 7x � 16 122.

��2, ��

x > �2

9x > �18

92x > �9

152 x � 4 > 3x � 5

123.

�3215, ��

x ≥ 3215

�152 x ≤ �16

20 � 8x ≤ 4 �12x

4�5 � 2x� ≤ 12�8 � x� 124.

��53, ��

x > �53

3x > �5

9 � 3x > 4 � 6x

12�3 � x� > 13�2 � 3x� 125.

��23, 17�

�23 < x ≤ 17

�2 < 3x ≤ 51

�19 < 3x � 17 ≤ 34

126.

��2, 10� �2 ≤ x < 10

�4 ≤ 2x < 20

�9 ≤ 2x � 5 < 15

�3 ≤2x � 5

3< 5 127.

��4, 4�

�4 ≤ x ≤ 4

�x� ≤ 4 128.

�1, 3�

1 < x < 3

�1 < x � 2 < 1

�x � 2� < 1

129.

or

or

���, �1� � �7, ��

x > 7 x < �1

x � 3 > 4 x � 3 < �4

�x � 3� > 4 130.

���, 0� � �3, ��

x ≤ 0 x ≥ 3

x �32 ≤ �

32 or x �

32 ≥ 3

2

�x �32� ≥ 3

2

131. If the side is 19.3 cm, then with the possible error of 0.5 cm we have:

353.44 cm2 ≤ area ≤ 392.04 cm2

18.8 ≤ side ≤ 19.8

132.

x > 36 units

33.33x > 1200

125.33x > 92x � 1200

133.

Critical numbers:

Test intervals:

Test: Is ?

By testing an x-value in each test interval in the inequality, we see that the solution set is: ��3, 9�

�x � 3��x � 9� < 0

���, �3�, ��3, 9�, �9, ��

x � �3, x � 9

�x � 3��x � 9� < 0

x2 � 6x � 27 < 0 134.

Critical numbers:

Test intervals:

Solution interval: ���, �1� � �3, ��

�3, �� ⇒ �x � 3��x � 1� > 0

��1, 3� ⇒ �x � 3��x � 1� < 0

���, �1� ⇒ �x � 3��x � 1� > 0

x � �1, x � 3

�x � 3��x � 1� ≥ 0

x2 � 2x � 3 ≥ 0

x2 � 2x ≥ 3

Page 121: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Review Exercises for Chapter 1 187

135.

Critical numbers:

Test intervals:

Test: Is

By testing an x-value in each test interval in the inequality, we see that the solution set is: ��4

3, 12�

�3x � 4��2x � 1� < 0?

���, �43, �, ��4

3, 12�, �12, ��

x � �43, x �

12

�3x � 4��2x � 1� < 0

6x2 � 5x � 4 < 0

6x2 � 5x < 4 136.

Critical numbers:

Test intervals:

Solution interval: ���, �3� � �52, ��

�52, �� ⇒ �2x � 5��x � 3� > 0

��3, 52� ⇒ �2x � 5��x � 3� < 0

���, �3� ⇒ �2x � 5��x � 3� > 0

x �52, x � �3

�2x � 5��x � 3� ≥ 0

2x2 � x � 15 ≥ 0

2x2 � x ≥ 15

137.

Critical numbers:

Test intervals:

Test: Is

By testing an value in each test interval in the inequality,we see that the solution set is: ��4, 0� � �4, ��.

x-

x�x � 4��x � 4� ≥ 0?

�4, ���0, 4�,��4, 0�,���, �4�,

x � 0, x � ±4

x�x � 4��x � 4� ≥ 0

x3 � 16x ≥ 0 138.

Critical numbers:

Test intervals:

Solution interval: ���, 0� � �0, 53��5

3, �� ⇒ 12x3 � 20x2 > 0

�0, 53� ⇒ 12x3 � 20x2 < 0

���, 0� ⇒ 12x3 � 20x2 < 0

x � 0, x �53

4x2�3x � 5� < 0

12x3 � 20x2 < 0

139.

Critical numbers:

Test intervals:

By testing an value in each test interval in the inequality,we see that the solution set is: ���, �5� � ��2, ��

x-

��2, ����5, �2�,���, �5�,

x � �2, x � �5

�x � 2x � 5

< 0

x � 8 � 2�x � 5�

x � 5< 0

x � 8x � 5

� 2 < 0 140.

Critical numbers:

Test intervals:

Solution interval: ���, 3� � �20, ��

�20, �� ⇒ 3x � 8x � 3

< 4

�3, 20� ⇒ 3x � 8x � 3

≥ 4

���, 3� ⇒ 3x � 8x � 3

< 4

x � 3, x � 20

x ≥ 20

3x � 8 ≤ 4x � 12

3x � 8x � 3

≤ 4

141.

Critical numbers:

Test intervals:

Test: Is

By testing an x-value in each test interval in the inequality,we see that the solution set is: ��5, �1� � �1, ��

��x � 5��x � 1��x � 1� ≤ 0?

���, �5�, ��5, �1�, ��1, 1�, �1, ��

x � �5, x � ±1

��x � 5�

�x � 1��x � 1�≤ 0

2x � 2 � 3x � 3

�x � 1��x � 1)≤ 0

2�x � 1� � 3�x � 1�

�x � 1��x � 1�≤ 0

2

x � 1≤

3

x � 1142.

Critical numbers:

Test intervals:

Solution intervals: ���, 3� � �5, ��

�5, �� ⇒ x � 53 � x

< 0

�3, 5� ⇒ x � 53 � x

> 0

���, 3� ⇒ x � 53 � x

< 0

x � 5, x � 3

x � 53 � x

< 0

Page 122: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

188 Chapter 1 Equations, Inequalities, and Mathematical Modeling

143.

Critical numbers:

Test intervals:

Test: Is

By testing an x-value in each test interval in the inequality,we see that the solution set is: ��4, �3� � �0, ��

�x � 4��x � 3�x

≥ 0?

���, �4�, ��4, �3�, ��3, 0�, �0, ��

x � �4, x � �3, x � 0

�x � 4��x � 3�x

≥ 0

x2 � 7x � 12

x≥ 0 144.

Critical numbers:

Test intervals:

Solution interval: ���, 0� � �2, ��

�2, �� ⇒ 1

x � 2�

1x

> 0

�0, 2� ⇒ 1x � 2

�1x

< 0

���, 0� ⇒ 1x � 2

�1x

> 0

x � 2, x � 0

1

x � 2�

1x

> 0

1

x � 2>

1x

145.

r > 4.9%

r > 0.0488

1 � r > 1.0488

�1 � r�2 > 1.1

5000�1 � r�2 > 5500 146.

t ≥ 9 days

�1000t ≤ �9000

10,000 � 2000t ≤ 1000 � 3000t

2000�5 � t� ≤ 1000�1 � 3t�

2000 ≤1000�1 � 3t�

5 � t

P �1000�1 � 3t�

5 � t

147. False

���18���2� � �36 � 6

��18��2 � ��18i���2i� � �36i2 � �6

148. False. The equation has no real solution. The solutions are

717650

±�3311i

650.

149. Rational equations, equations involving radicals, andabsolute value equations, may have “solutions” that are extraneous. So checking solutions, in the originalequations, is crucial to eliminate these extraneous values.

150. Sample answer: The first equivalent inequality written isincorrect. It should be This leads tothe solution or �2, ��.���, �30

11�11x � 4 ≤ �26.

Problem Solving for Chapter 1

1.

Time (in seconds)

Dis

tanc

e (i

n fe

et)

y

x

Page 123: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Problem Solving for Chapter 1 189

2. (a)

(b)

When

When

When n � 10: 12�10��11� � 55

n � 8: 12�8��9� � 36

n � 5: 12�5��6� � 15

1 � 2 � 3 � . . . � n �12n�n � 1�

1 � 2 � 3 � 4 � 5 � 6 � 7 � 8 � 9 � 10 � 55

1 � 2 � 3 � 4 � 5 � 6 � 7 � 8 � 36

1 � 2 � 3 � 4 � 5 � 15 (c)

or

Since n is a natural number, choose n � 20.

n � 20n � �21

�n � 21��n � 20� � 0

n2 � n � 420 � 0

n�n � 1� � 420

12n�n � 1� � 210

3. (a)

thus:

(b)

(c)

or a � 7.877a � 12.123

� 10 ±10����� � 3�

�20� ± 20���� � 3�

2�

�20� ± �400�2 � 1200�

2�

a �20� ± ���20��2 � 4��300�

2�

�a2 � 20�a � 300 � 0

300 � 20�a � �a2

300 � �a�20 � a�

A � �a�20 � a�

a � b � 20 ⇒ b � 20 � a,

A � �ab (d)

(e) The a-intercepts occur at and Both yield anarea of 0. When and you have a verticalline of length 40. Likewise when and youhave a horizontal line of length 40. They represent theminimum and maximum values of a.

(f) The maximum value of A is This occurswhen and the ellipse is actually a circle.a � b � 10

100� � 314.159.

a � 20, b � 0a � 0, b � 20

a � 20.a � 0

00 20

350

A � na�20 � a�

a 4 7 10 13 16

A 64 91 100 91 64�����

4.

(a)

miles per hour.

(b)

miles per hour.

No, actually it can survive wind blowing at timesthe speed found in part (a).

(c) The wind speed in the formula is squared, so a smallincrease in wind speed could have potentially seriouseffects on a building.

�2

s � 125

s2 � 15625

0.00256s2 � 40

s � 88.4

s2 � 7812.5

0.00256s2 � 20

P � 0.00256s2 5.

(a)

(b)

seconds

(c) The speed at which the water drains decreases as theamount of the water in the bathtub decreases.

t �225�12.5

��3� 146.2

0 � ��12.5 ���3225

t�2

t �225

��3�5 � �12.5 � � 60.6 seconds

�12.5 � 5 ���3225

t

12.5 � �5 ���3225

t�2

h � �5 �8��31800

t�2

� �5 ���3225

t�2

l � 60�, w � 30�, h0 � 25�, d � 2�

h � ��h0 �2�d 2�3

lw t�2

Page 124: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

190 Chapter 1 Equations, Inequalities, and Mathematical Modeling

6. (a) If then and

(b)

There is no integer m such that equals a negativenumber. cannot be factored over the integers.x2 � 9

m2

m��m� � 9 ⇒ �m2 � 9 ⇒ m2 � �9

m � n � 0 ⇒ n � �m

m � n � 0.mn � 9x2 � 9 � �x � m��x � n� 7. (a) 5, 12, and 13; 8, 15, and 17

7, 24, and 25

(b) which is divisible by 3, 4, and 5

which is divisible by 3, 4, and 5

which is also divisible by 3, 4,and 5

(c) Conjecture: If where a, b, and c arepositive integers, then is divisible by 60.abc

a2 � b2 � c2

7 � 24 � 25 � 4200

8 � 15 � 17 � 2040

5 � 12 � 13 � 780

8.Equation

(a)1

(b)

(c)

(d)x � 5 ± 3i

34105 � 3i, 5 � 3ix2 � 10x � 34 � 0

�2x � 3��2x � 3� � 0 �940�

32, 3

2

4x2 � 9 � 0

�2x � 1��x � 3� � 0 �32�

52

12, �3

2x2 � 5x � 3 � 0

�x � 2��x � 3� � 0�6�2, 3

x2 � x � 6 � 0

x1 � x2x1 � x2x1, x29.

x1 � x2 �ca

x1 � x2 � �ba

x2 �ba

x �ca

� 0

ax2 � bx � c � 0

10. (a) (i)

(ii) ��5 � 5�3i2 �3

� 125

��5 � 5�3i2 �3

� 125 (b) (i)

(ii) ��3 � 3�3i2 �3

� 27

��3 � 3�3i2 �3

� 27 (c) (i) The cube roots of 1 are:

(ii) The cube roots of 8 are:

(iii) The cube roots of 64 are:4, �2 ± 2�3i

2, �1 ± �3i

1, �1 ± �3i

2

12.

Since a and b are real numbers, is also a real number.a2 � b2

�a � bi��a � bi� � a2 � abi � abi � b2i2 � a2 � b2

11. (a)

�1 � i

2�

12

�12

i

�1

1 � i�

11 � i

�1 � i1 � i

zm �1z

(b)

�3 � i

10�

310

�1

10i

�1

3 � i�

13 � i

�3 � i3 � i

zm �1z

(c)

��2 � 8i

68� �

134

�2

17i

�1

�2 � 8i�

�2 � 8i�2 � 8i

�1

�2 � 8i

zm �1z

Page 125: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Problem Solving for Chapter 1 191

13. (a)

The terms are:

The sequence is bounded so is in the Mandelbrot Set.

(b)

The terms are:

The sequence is unbounded so is not in the Mandelbrot Set.

(c)

The terms are:

The sequence is bounded so is in the Mandelbrot Set.c � �2

�2, 2, 2, 2, 2, . . .

c � �2

c � 1 � i

1 � i, 1 � 3i, �7 � 7i, 1 � 97i, �9407 � 193i, . . .

c � 1 � i

c � i

i, �1 � i, �i, �1 � i, �i, �1 � i, �i,�1 � i, �i, . . .

c � i

14.

This equation will have two solutions when or when

This equation will have one solution when or when

This equation will have no solutions when or when k > 2.1 �k2

< 0

k � 2.1 �k2

� 0

k < 2.1 �k2

> 0

��x � 1�2 � 1 �k2

x � 2�x � 1 � 1 �k2

x � 2�x � �k2

2x � 4�x � k � 0 Complete the square.

4�x � 2x � k Number of solutions (real) Some k-values

2

1 2 only

0 3, 4, 5

�1, 0, 1

15.

From the graph we see that on the intervals ���, �2� � ��1, 2� � �2, ��.

x4 � x3 � 6x2 � 4x � 8 > 0

−5

−5 5

10

� �x � 2�2�x � 1��x � 2�

y � x4 � x3 � 6x2 � 4x � 8 16.

The greatest amount you could expect to get is 66 ounces and the least amount is 62 ounces.

4�15.5� � 62, 4�16.5� � 66

15.5 ≤ w ≤ 16.5

�12 ≤ w � 16 ≤ 1

2

w � 16 ≤ 12

Page 126: CHAPTER 1 Equations, Inequalities, and Mathematical ModelingSection 1.1 Graphs of Equations 69 4. (a) Yes, the point is on the graph. 16 3 16 3 8 3 24 3? 16 3 8 3 8 16 3 1 3 8 2 14

Practice Test for Chapter 1

192 Chapter 1 Equations, Inequalities, and Mathematical Modeling

1. Graph 3x � 5y � 15. 2. Graph y � �9 � x.

3. Solve 5x � 4 � 7x � 8. 4. Solve x

3� 5 �

x

5� 1.

5. Solve 3x � 1

6x � 7�

2

5. 6. Solve �x � 3�2 � 4 � �x � 1�2.

7. Solve A �12�a � b�h for a. 8. 301 is what percent of 4300?

9. Cindy has in quarters and nickels. How many of each coin does she have if there are 53 coins in all?$6.05

10. Ed has $15,000 invested in two funds paying and 11% simple interest, respectively. How much is invested in each if the yearly interest is $1582.50?

912%

11. Solve by factoring.28 � 5x � 3x2 � 0

12. Solve by taking the square root of both sides.�x � 2�2 � 24

13. Solve by completing the square.x2 � 4x � 9 � 0

14. Solve by the Quadratic Formula.x2 � 5x � 1 � 0

15. Solve by the Quadratic Formula.3x2 � 2x � 4 � 0

16. The perimeter of a rectangle is 1100 feet. Find the dimensions so that the enclosed area will be 60,000 square feet.

17. Find two consecutive even positive integers whose product is 624.

18. Solve by factoring.x3 � 10x2 � 24x � 0 19. Solve 3�6 � x � 4.

20. Solve �x2 � 8�2�5 � 4. 21. Solve x4 � x2 � 12 � 0.

22. Solve 4 � 3x > 16. 23. Solve �x � 3

2 � < 5.

24. Solve x � 1

x � 3< 2. 25. Solve �3x � 4� ≥ 9.