appendix a x appendix a.1 (page a8) -...

A318 Answers to All Exercises and Tests

Appendix AAppendix A.1 (page A8)

Vocabulary Check (page A8)1. rational 2. irrational 3. absolute value4. composite 5. prime 6. variables; constants7. terms 8. coefficient 9. Zero-Factor Property

1. (a) 5, 1, 2 (b) 0, 5, 1, 2(c)(d) (e)

2. (a) 12, 5 (b) 12, 5, 0 (c)(d) (e)

3. (a) 1 (b) 1 (c)(d)(e)

4. (a) 4 (b) 4 (c)(d)(e)

5. (a) (b) (c)

(d) (e)6. (a) (b)

(c)

(d) (e)

7. 0.625 8. 9. 10.11. 12.13.

14.

15.

16.

17.

18.

19. (a) denotes the set of all real numbers less than orequal to 5.

(b) (c) Unbounded

20. (a) denotes the set of all real numbers greater thanor equal to

(b) (c) Unbounded

21. (a) denotes the set of all real numbers less than 0.(b) (c) Unbounded

22. (a) denotes the set of all real numbers greater than 3.(b) (c) Unbounded

23. (a) denotes the set of all real numbers greater thanor equal to 4.

(b) (c) Unbounded

24. (a) denotes the set of all real numbers less than 2.(b) (c) Unbounded

25. (a) denotes the set of all real numbers greaterthan and less than 2.

(b) (c) Bounded

26. (a) denotes the set of all real numbers greaterthan or equal to zero and less than or equal to 5.

(b) (c) Bounded

27. (a) denotes the set of all real numbers greaterthan or equal to and less than 0.

(b) (c) Bounded

28. (a) denotes the set of positive real numbers lessthan or equal to 6.

(b) (c) Bounded

29. (a) denotes the set of all real numbers greater thanor equal to and less than 5.

(b) (c) Bounded

30. (a) denotes all real numbers greater than andless than or equal to 2.

(b) (c) Bounded

31. 32. 33.34. 35. 36.37. 38. 39. 10 40. 041. 5 42. 3 43. 44. 45.46. 47. 48. 1 49.50. 51.52. 53. ��2� � ��2��6� < ��6�

�5 � ��5��4� � �4��3� > ��3��1�9

�1�6�12.5% ≤ r ≤ 5%W > 65

�3 ≤ k < 510 ≤ t ≤ 22y ≤ 25y ≥ 0�6 ≤ y < 0�2 < x ≤ 4

x210−2 −1

�1��1, 2�0 1 2 3 4 5 6−1−2

x

�2��2, 5�

x0 1 2 3 4 5 6

0 < x ≤ 6

0−1

x

�1�1 ≤ x < 0

x

0 1 2 3 4 5

0 ≤ x ≤ 5

−2 −1 0 1 2

x

�2�2 < x < 2

x

0 1 2 3 4

��, 2�1 2 3 4 5 6 7

x

�4, ��

x0 1 2 3 4 5 6

x > 3

−2 −1 0 1 2

x

x < 0

x

−1−2−3−4 0

�2.x ≥ �2

0 1 2 3 4 5 6

x

x ≤ 5�

87 < �

37

−1 0−2

− 37− 8

7

56 > 2

3

0 1

32

65

1 < 163

0 1 2 3 4 5 6

163

32 < 7

1 2 3 4 5 6 7

23

�3.5 < 1

10−1−2−3−4

−3.5

�4 > �8−8 −7 −6 −5 −4

�6 < �2.5�1 < 2.50.540.1230.3

�

2 �

12

5, �9, 3.12, 25, �17, �11.1, 13

25, �17, �9, 7, 1325, �9, 7, 1325, �9, 7, 13

��, 12�2�13, 63, �7.5, �1, 8, �22

63, �1, 8, �226

3, 863, 82.3030030003 . . . , �10�75, 4, �4.63, 0.7575

�75, 40.010110111 . . .2.01, �13, 1, �6, 0.666 . . .

�13, 1, �6�5�7, �7

3, 0, 3.12, 54, �3, 12, 5�7, 12, 5, 0, �3

�2�72, 23, �9, 5, 0, 1, �4, 2, �11

�9, 5, 0, 1, �4, 2, �11

333202_App_AN.qxd 12/9/05 10:20 AM Page A318

Answers to All Exercises and Tests A319

AP

PE

ND

IX A

54. 55. 51 56. 57.58. 59. 60. 14.9961.

Because the actual expenses differ from the budget by morethan $500, there is failure to meet the “budget variancetest.”

62.

Because the difference between the actual expenses and thebudget is less than $500 and less than 5% of the budgetedamount, there is compliance with the “budget variancetest.”

63.

Because the difference between the actual expenses and thebudget is less than and less than of the budgetedamount, there is compliance with the “budget variancetest.”

64.

Because the difference between the actual expenses and thebudget is less than $500 and less than 5% of the budgetedamount, there is compliance with the “budget variancetest.”

65. (a)

(b)

66.

67. 68. 69.70. 71.72.73. and 4 are the terms; 7 is the coefficient.74. are the terms; 6 and are the coefficients.75. are the terms; and are the

coefficients.76. and 1 are the terms; is the coefficient.77. are the terms; 4 and are the

coefficients.78. and are the terms; 3 and are the coefficients.79. (a) (b) 80. (a) 30 (b)81. (a) (b) 82. (a) (b) 083. (a) Division by is undefined. (b)84. (a) (b) Division by 0 is undefined.85. Commutative Property of Addition86. Multiplicative Inverse Property87. Multiplicative Inverse Property88. Additive Inverse Property89. Distributive Property90. Additive Identity Property91. Multiplicative Identity Property92. Distributive Property93. Associative Property of Addition94. Associative and Commutative Properties of Multiplication95. Distributive Property96. Associative Property of Multiplication

Multiplicative Inverse Property

Multiplicative Identity Property

97. 98. 99. 100. 101. 48

102. 103. 104.

105. (a)

(b) The value of approaches infinity as n approaches 0.106. (a)

(b) The value of approaches 0 as n increases withoutbound.

5�n

5�n

5x27

5x12

�3

5966

38

27

12

� 12 � 1 � 12

17�7 � 12� � �17 � 7�12

12

00�10214

�12�6�10�

14�x2�43x 4

124x3, x�2, and �5

3�33�3x2

�8�3�3x2, �8x, and �11�56x3 and �5x

7x�48� � 82�� 34�

�326 � 351� � 25 miles�y � a� ≤ 2�y� ≥ 6�x � 10� ≥ 6�x � 5� ≤ 3

Under 358.64%

65 and older38.19%

35−4412.84%

45−5418.21%

55−6422.11%

Surp

lus

or d

efic

it (

in b

illio

ns)

240

192

144

96

48

0.3

(s)

2.8

(d)

73.8

(d)

221.

2 (d

)

236.

4 (s

)

Year

1960

1970

1980

1990

2000

0.05�$2575� � $128.75�$2613 � $2575� � $38 < $500

5%$500

0.05�$37,640� � $1882�$37,335 � $37,640� � $305 < $500

0.05�$9400� � $470�$9772 � $9400� � $372 < $500

0.05�$112,700� � $5635�$113,356 � $112,700� � $656 > $500

12875

52

5251��2� > �2

n 1 0.5 0.01 0.0001 0.000001

5 10 500 50,000 5,000,0005�n

Year Expenditures Surplus or deficit(in billions) (in billions)

1960 $92.2 $0.3 (s)

1970 $195.6 $2.8 (d)

1980 $590.9 $73.8 (d)

1990 $1253.2 $221.2 (d)

2000 $1788.8 $236.4 (s)

n 1 10 100 10,000 100,000

5 0.5 0.05 0.0005 0.000055�n



107. False. If then where

108. False. The denominators cannot be added when addingfractions.

109. (a) No. If one variable is negative and the other ispositive, the expressions are unequal.

(b)The expressions are equal when u and v have the samesign. If u and v differ in sign, is less than

110. Yes. y is nonnegative if y is positive if 111. The only even prime number is 2, because its only factors

are itself and 1.112.

113. (a) Negative (b) Negative114. (a) Positive (b) Positive115. Yes.

Appendix A.2 (page A20)

Vocabulary Check (page A20)1. exponent; base 2. scientific notation3. square root 4. principle nth root5. index; radicand 6. simplest form7. conjugates 8. rationalizing9. power; index

1.2.3. 4. 5. (a) 27 (b) 816. (a) 125 (b) 7. (a) 1 (b)8. (a) 5184 (b) 9. (a) (b)

10. (a) (b) 1 11. (a) (b) 4

12. (a) (b) 13. 14. 0.244

15. 2.125 16. 5184 17. 18. 19. 6

20. 21. 22. 23. 1

24. 25. (a) (b)

26. (a) (b) 1 27. (b)

28. (a) (b) 29. (a) (b)

30. (a) (b) 31. (a) 1 (b)

32. (a) 1 (b) 33. (a) (b)

34. (a) (b) 35. (a) (b)

36. (a) (b) 1 37. square miles

38. kilometers39. gram per cubic centimeter40. inch41. 4,568,000,000 ounces 42. 15,000,00043. 0.00000000000000000016022 coulomb44. 0.00009 meter 45. (a) 50,000 (b) 200,00046. (a) 60,000 (b)47. (a) 954.448 (b)48. (a) (b) 1.47949. (a) 67,082.039 (b) 39.79150. (a) 0.064 (b) 0.03 51. (a) 3 (b)52. (a) 3 (b) 216 53. (a) (b)54. (a) (b) 55. (a) (b) 256. (a) (b) 57. (a) 7.550 (b)58. (a) 12.651 (b) 2.23659. (a) (b) 0.00560. (a) 3.495 (b) 5.906 61. (a) 4 (b)62. (a) 6 (b) 63. (a) (b)

64. (a) (b) 65. (a) (b)

66. (a) (b)

67. (a) (b)

68. (a) (b)69. (a) (b) 70. (a) (b)71. (a) (b)72. (a) (b)73. (a) (b)74. (a) (b)

75. 76.

77. 78. 79.

80. 81. 82.

83. 84. 85.

86. 87. 88.

89. 90. 91. ��216�1�3��1445�32

641�391�2�1

2��7 � 3�

2

3��5 � �3�2

3�2

2�2

3��6 � �5�5 � �311

�10

2

�33

5 � �32 � 425 > �32 � 22

� 3

11�

�3�11

�5 � �3 > �5 � 3

50x�5x4�x3 � 718�5x13�x � 1

23�x��3�84�x4�y2�x

11 3�27�322�234�22x 5�5x3z44�3�x��y

5�x��3

y 22x 3�2x 2

4a2

�b��23y 2�6x

18�zz26x�2x

5�3

2

2 3�2

3

3 3�22�23x2

2 5�3x�0.011

�7.225�625�35

�423

11000

278

18

32

4.907 � 1017

3.077 � 1010

2.0 � 1011

�C3.937 � 10�5

8.99 � 10�5

9.46 � 1012

5.73 � 1071

x

b5

a 533n

125x 9

y1232y 2

10

x�2x3

1

�z � 2� 4

1

4x 4

5184

y7

1

r 2

4

3�x � y�2

7

x

5

2y4�3z7

3x224y29x2

5x6�125z3�527

�48�54�135

716�24

�1600181

712

56

163

�124364�

35

�919

��10�54.96

��2� � ��2� � ��2� � ��2� � ��2� � ��2� � ��2�8 � 8 � 8 � 8 � 8

�a� � �a if a < 0.

Realnumbers

Irrationalnumbers

Integers. . . , 3, 2, 1,

0, 1, 2, 3, . . .− − −

Nonintegerfractions

(positive andnegative)

Rationalnumbers

Wholenumbers

0, 1, 2, . . .

Naturalnumbers

1, 2, 3, . . .

Zero

Negativeintegers

. . . , 3, 2, 1,− − −

y > 0.y ≥ 0.�u� � �v�.

�u � v�

�u � v� ≤ �u� � �v�

a b 0.1a

>1b

,a < b,



AP

PE

ND

IX A

92. 93. 94.

95. 96. 97. , 98. x

99. (a) (b) 100. (a) (b)101. (a) (b)102. (a) (b)

103. 104. 0.026 inch

105. (a)

(b)106. 8.32 minutes107. True. When dividing variables, you subtract exponents.108. False. When a power is raised to a power, you multiply the

exponents:

109. using the property

110. (a) 3 is also raised to the power, so

(b) When two powers have the same base, the exponentsare added:

(c) When a power is raised to a power, the exponents aremultiplied:

(d) The square of a binomial contains a cross-productterm:

(e) If but (f ) Radicals can be added together only if they have the

same radicand and index:111. When any positive integer is squared, the units digit is 0,

1, 4, 5, 6, or 9. Therefore, is not an integer.112. No. Rationalizing the denominator produces a number

equivalent to the original fraction; squaring does not.


Vocabulary Check (page A31)1. 2. descending3. monomial; binomial; trinomial 4. like terms5. First terms; Outer terms; Inner terms; Last terms6. factoring 7. completely factored

1. d 2. e 3. b 4. a 5. f 6. c7. 8.

9. 10.11. (a)

(b) Degree: 5; Leading coefficient:(c) Binomial

12. (a)(b) Degree: 2; Leading coefficient: 2(c) Trinomial

13. (a)(b) Degree: 4; Leading coefficient:(c) Trinomial

14. (a)(b) Degree: 1; Leading coefficient: 7(c) Monomial

15. (a)(b) Degree: 5; Leading coefficient: 1(c) Binomial

16. (a)(b) Degree: 2; Leading coefficient: 25(c) Trinomial

17. (a) 3(b) Degree: 0; Leading coefficient: 3(c) Monomial




21. (a)(b) Degree: 3; Leading coefficient: 4(c) Monomial


23. Polynomial:24. Not a polynomial because of the negative exponent25. Not a polynomial because it includes a term with a negative

exponent26. Polynomial:27. Polynomial:28. Not a polynomial because of the square root29. 30. 31.32. 33.34. 35.36. 37.38. 39.40. 41. 42.43. 44. 45. �

12x2 � 12x�7y4 � 4y37.5x3 � 9x

4x 4 � 12x�4x 4 � 4x�15x2 � 6x�15z 2 � 5z4y 4 � 2y 3 � 3y 2

3x3 � 6x 2 � 3xy 3 � y 2 � 3y � 712z � 81.3x 4 � 8.4x � 34.1

8.3x3 � 29.7x2 � 11�2x2 � 43x3 � 2x � 2x2 � 2x�2x � 10

�y 4 � y 3 � y 2

12x2 � x �

32

�3x3 � 2x � 8

�1�x5y � 2x2y2 � xy4

4x3y

2x � 3

�4�4x5 � 6x4 � 1

t2 � 9

25y2 � y � 1

x5 � 1

7x

�3�3x4 � 2x2 � 5

2x2 � x � 1

�12

�12x5 � 14x

20x3 � 5�15x 4 � 1

6x5 � 3x � 1�2x3 � 4x2 � 3x � 20

n; an; a0

�5233

�2 � �2 � 2�2.

�4x2 � 2�x�.2x < 0:x < 0, then �4x2 > 0�a � b�2 � a2 � 2ab � b2.

�a2b3�4 � a8b12.

y3 � y2 � y5.

�3x��1 �13x

.�1

am

am � am�m � a0 � 1.

am

an � am�n:a0 � 1, a � 0,

�an�k � ank.

t → 8.64�3 � 14.96

�

2� 1.57 seconds

a 6�10ab3 4�3�x � 1�8�2x2 4�2

3x2�x3��x � 1�2�3

x > 01

x 3xy1�3

2

x

4�165813�45��243

h 0 1 2 3 4 5 6

t 0 2.93 5.48 7.67 9.53 11.08 12.32

h 7 8 9 10 11 12

t 13.29 14.00 14.50 14.80 14.93 14.96

333202_App_AN.qxd 12/12/05 2:11 PM Page A321


46. 47.48. 49.50. 51.52. 53. 54.55. 56. 57.58. 59.60. 61.62. 63.64. 65.66. 67.68.69.70. 71.72. 73.74. 75. 76.77. 78.79. 80. 81.82. 83. 84.85. 86. 87.88. 89. 90.91. 92.93. 94.95. 96. 97.98. 99.

100. 101.102. 103.104. 105.

106. 107.108. 109.110. 111.112. 113.114. 115. 116.117. 118. 119.120. 121. 122.123. 124.125.126. 127.128.129.130. 131.132. 133.134. 135.136. 137.138. 139.140. 141.142. 143.144. 145.146. 147.148. 149.150.151. 152.

153. 154.155. 156.157. 158.159. 160.161. 162. 163.164. 165.166. 167.168. 169.170. 171.172. 173.174. 175.176. 177.178. 179.180.181.182.183. 184.185. 186.187.

188. 189.190.191.

192.

193. 194.195.196.197. Two possible answers: 2,198. Two possible answers: 3,199. Two possible answers:200. Two possible answers: 9, 7201. (a) (b) $85,000202. (a) (b) $548203. (a)

(b)

(c) The amount increases with increasing r.204. (a)

(b)

(c) The amount increases with increasing r.

1200r 3 � 3600r 2 � 3600r � 1200

500r 2 � 1000r � 500P � 24x � 460P � 22x � 25,000

�2, �4�8�12

�25, 25, �14, 14, �11, 11, �10, 10�11, 11, �4, 4, �1, 1

�51, 51, �15, 15, �27, 27�14, 14, �2, 2

��x2 � 1� 4x 2

2� 1

3�x6 � 1�4�3x � 2�2�33x6 � 20x5 � 3��2x�x � 5�3�x � 5�

�x � 2�2�x � 1�3�7x � 5�7�x2 � 1��3x2 � 1�5�1 � x�2�3x � 2��4x � 3�

��x � 1��x � 3��x � 9��3 � 4x��23 � 60x��2x � 1��6x � 1�5�x � 2��x2 � 2x � 4�

2�t � 2��t 2 � 2t � 4��x � 2��x � 4��x � 2��x � 4��x � 1�2 �x � 1� 2

�t � 6��t � 8�15�x2 � 5��x � 5�

14�x2 � 3��x � 12��u � 2��3 � u 2�x�x � 4��x2 � 1��5 � x��1 � x2�

�3x � 1��x2 � 5�196 �4x � 3��3x � 2�

181�x � 36��x � 18��5x � 3)�x � 2�

�9x � 1��x � 1�y�2y � 3��y � 5��2x�x � 1��x � 2��3x � 1��3x � 1�

�1 � 2x�2�8 � x��2 � x��x � 1�2x�x � 3��x � 3�x2�x � 4�

12�x � 2��x � 2�6�x � 3��x � 3��x � 1��12x � 1��3x � 1��5x � 2��2x � 3��3x � 5��2x � 1��3x � 2�

�x � 3��2x � 3��x � 2��3x � 4�

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

�3 � x��2 � x3��x � 2��5x2 � 3��2x � 1��x2 � 3��x � 5��x2 � 5�

�x � 1��x2 � 2��5u � 2��u � 3��3z � 2��3z � 1��3x � 1��4x � 1�

�5x � 1��x � 5��2x � 1��x � 1��3x � 2��x � 1��x � 6��x � 7�

�x � 20��x � 10��z � 8��z � 3��y � 5��y � 4��t � 2��t � 3��s � 3��s � 2��x � 2��x � 3�

�x � 2��x � 1��4x � y��16x2 � 4xy � y2��u � 3v��u2 � 3uv � 9v2��3x � 2��9x2 � 6x � 4�

�2t � 1��4t 2 � 2t � 1��z � 5��z2 � 5z � 25��y � 4��y 2 � 4y � 16�

�x � 3��x2 � 3x � 9��x � 2��x2 � 2x � 4��z �

12�2�x �

23�2�2x � y�2

�3u � 4v�2�6y � 9�2�5y � 1�2

�3x � 2�2�2t � 1�2�x � 5�2

�x � 2�2�5x � 4y��5x � 4y��3u � 2v��3u � 2v��z �z � 10�

�x � 1��x � 3��25y � 8��2

5y � 8��4x �

13��4x �

13��4�3y � 1��3y � 1�

2�4y � 3��4y � 3��x � 7��x � 7��x � 9��x � 9�2

5�y � 1��2y � 5�

23�x � 6��x � 3�1

3y�y3 � 15y � 6�

12x�x2 � 4x � 10�1

3 �y � 15�

12�x � 8�3x�3x � 1��x � 3��x � 1�

�x � 2��3x � 4��x � 1��x � 6�2x�2x 2 � 3x � 6�2x�x 2 � 3�

5�y � 6�3�x � 2�x2 � 2x�3 � 3x2 � 2�5x � 525 � xx � y

x4 � y 4u4 � 162x 2 � 8x � 62x2 � 2x6.25y2 � 92.25x2 � 16

2.25y2 � 9y � 91.44x2 � 7.2x � 94x2 �

125

19x2 � 44

9t2 �203 t � 25

14x2 � 3x � 99a 6 � 16b4

4r 4 � 25x2 � 2xy � y 2 � 2x � 2y � 1x2 � 2xy � y2 � 6x � 6y � 9x2 � 2xy � y 2 � 1

m 2 � n 2 � 6m � 964x2 � 48x � 916x6 � 24x3 � 927x3 � 54x 2y � 36xy 2 � 8y 3

8x3 � 12x 2y � 6xy 2 � y 3x3 � 6x2 � 12x � 8x3 � 3x 2 � 3x � 164x2 � 80x � 254x 2 � 20xy � 25y 216x2 � 40x � 25

4x 2 � 12x � 94x2 � 9y 2x2 � 4y 2

4x2 � 9x2 � 100x 4 � 13x 2 � 4x 4 � x 2 � 128x 2 � 29x � 6

6x 2 � 7x � 5x 2 � 5x � 50x 2 � 7x � 12�

74 y2 � 8y

r % 3% 4%

$525.31 $530.45 $540.80500�1 � r�2

212

r % 5%

$546.01 $551.25500�1 � r�2

412

r 2% 3%

$1273.45 $1311.27 $1330.461200�1 � r�3

312%

r 4%

$1349.84 $1369.401200�1 � r�3

412%



AP

PE

ND

IX A

205. (a)(b)

206. (a)(b)

207. 208.209. (a) (b)210. (a)

(b)

(c) The stopping distance increases at an accelerating rateas speed increases.

211.

212.

213.

214.

215. 216.217. 218.

219. (a) (b)

220.221. False.222.

223. True. 224. False. A perfect square trinomial can be factored as the

binomial sum squared.225. 226. n 227.

228.If either x or y is zero, then

229.230.231. is completely factored.232. Answers will vary. Sample answer: The roots of the equa-

tion can be found when a polynomial is in factored form.233. Answers will vary. Sample answer:


Vocabulary Check (page A42)1. domain 2. rational expression 3. complex4. smaller 5. equivalent 6. difference quotient

1. All real numbers 2. All real numbers3. All nonnegative real numbers4. All positive real numbers5. All real numbers x such that 6. All real numbers x such that 7. All real numbers x such that 8. All real numbers x such that 9. 10.

11. 12. 13.

14. 15. 16.

17. 18. 19.

20. 21.

22. 23.

24. 25.��x2 � 1�

�x � 2� , x 2�x � 6��x � 1��x � 10��x � 1�

y � 4

y � 6, y 3

x � 2

x � 1, x �10

x�x � 3�x � 2

, x �2��x � 5�, x 5

y � 4, y �4�4, x 3�1

2, x 5

9x

2, x �1

�4y5

, y 12

2x2

x � 1

3y

y � 1, x 0

3

10y 3

3x

2, x 0

�x � 1�, x �13x, x 0x ≤ 6x ≥ �1x �

12

x 2

x2 � 3

x3n � y2n

�xn � yn��x2n � xnyn � y2n��xn � yn��xn � yn�

�x � y�2 � x2 � y2.�3 � 4�2 � 49 25 � 32 � 4 2.

�x 3 � 8x 2 � 2x � 7m � n

a2 � b2 � �a � b��a � b� � 9 � 3 � 6 � 4x � 3 � 4x � 6

False. �4x � 3� � ��4x � 6��4x2 � 1��3x � 1� � 12x3 � 4x2 � 3x � 1

kx�Q � x�

V � 2��R � r

2 �R � r��h�h�R � r��R � r�

58�x � 7��x � 1�4�6 � x��6 � x�

r 2�4 � ��4� �r � 1�

x

x x x x

x

x

1

1 11 1

11 1

1

x

x x x x x x x

x

x

x

1

1 1 11 1 1

1 1 1 1 1 11

x

x x

1

x

x x x x x

x

x

1 1 1 1

11

1

1 11 1

1

x

x x x x x x x

x x

x

x

1

1 11 1

11

1 1 1 1 1 11

x x

x x x

T � 0.0475x 2 � 1.099x � 0.2330x 23x 2 � 8x2x2 � 46x � 25244x � 308

V �12�6x3 � 135x2 � 675x�

V � 4x3 � 88x2 � 468x

x (cm) 1 2 3

V 384 616 720�cm3�

x (cm) 3 5 7

V 486 375 84�cm3�

30 40 55

T (feet) 75.95 120.19 204.36

x �mi�hr�



26. 27.

28.

29.

The expressions are equivalent except at 30.

The expressions are equivalent except at 31. The expression cannot be simplified.32. The polynomial, can not be factored.

33. 34.

35. 36.

37. 38.

39. 40.

41. 42.

43. 44. 45. 46.

47. 48.

49. 50.

51. 52.

53. The error was incorrect subtraction in the numerator.54. The error was an incorrect expansion of in the

numerator.

55. 56.

57. 58.

59. 60. 61.

62. 63. 64.

65. 66.

67. 68.

69. 70.

71.

72. 73.

74. 75.

76.

77. 78.

79. (a) minute (b) minute(s) (c) minutes

80. 81. (a) 9.09% (b) 9.09%

82. (a) 4.57% (b) 4.57%

83. (a)

(b) The model is approaching a T-value of 40.84. (a)

(b) The estimates are fairly close to the actual numbers ofhouseholds.

(c)

Rate ��0.2151t3 � 2.408t2 � 7.75t � 5.5

�0.00146t4 � 0.0403t3 � 0.491t2 � 23.81t � 0.3

288�MN � P�N�MN � 12P�;

288�MN � P�N�MN � 12P�;

8t15

6016

�154

x

16

1

16

8�x � 2��x � 4�2x 0

x2�2x � 1�,

h 01

�x � h � 2 � �x � 2,

h 01

�x � h � 1 � �x � 1,

�1

�z � 3 � �z

1�x � 2 � �x

h 01

�x � h � 1��x � 1�,

h 0�1

�x � 4��x � h � 4�,

h 0�2x � h

x2�x � h�2,h 0�1

x�x � h�,

x2 � 2

x3 �1 � x2�1�2

3x � 1

3, x 0

4x3�2x � 1�2 � 2x�2x � 1�1�2

2x3 � 2x2 � 5�x � 1�1�2

�2x�x � 5��x � 5�4

�1�x2 � 1�5

x8 � 5x3

x7 � 2x2

�1t2�t2 � 1

2x � 12x

, x > 0

x � 1

x � 1, x 0x�x � 1�, x �1, 0

4x

x � 4, x 0, 4

1

2, x 2

�x � 2�2

4x � 1

�x � 1��x � 1�2 � x

x2 � 1, x 0

6�2x � 3��x � 2��x � 1��x � 4�

�x2 � 3

�x � 1��x � 2��x � 3�

2x � 5

x � 5�

2

x � 2

8 � 5x

x � 1

6x � 13

x � 3

x

x � 3

x � 5

x � 1

13

, x ±7�x � 6��x � 1�

x2 , x 6

x � y

x�x � y� 2, x �2y

t � 3

�t � 3��t � 2�, t �2

�8

5, y �3, 4

r � 1

r, r 1

�x � 13

5x2, x 3

1

5�x � 2�, x 1

x � 5

4�2x � 3�, x �5

�

4, r 0

x2 � 25,

x � 3.

x � 3.

y �y � 3�y 2 � y � 1

, y �1

z � 21

x � 1, x ± 3

x 0 1 2 3 4 5 6

1 2 3 Undef. 5 6 7

1 2 3 4 5 6 7x � 1

x2 � 2x � 3

x � 3

x 0 1 2 3 4 5 6

Undef.

18

17

16

15

14

13

12

1

x � 2

18

17

16

14

13

12

x � 3

x2 � x � 6

t 0 2 4 6 8 10

T 75 55.9 48.3 45 43.3 42.3

t 12 14 16 18 20 22

T 41.7 41.3 41.1 40.9 40.7 40.6

Year, t Banking Paying bills(millions) (millions)

2002 21.9 13.9

2003 27.0 17.8

2004 31.4 21.5

2005 35.6 25.0

2006 40.2 28.1

2007 45.6 31.0



AP

PE

ND

IX A

(d)

Explanation will vary.85. False. In order for the simplified expression to be

equivalent to the original expression, the domain of thesimplified expression needs to be restricted. If n is even,

If n is odd,86. False. The two expressions are equivalent for all values of

x such that 87. Completely factor each polynomial in the numerator and in

the denominator. Then conclude that there are no commonfactors.


Vocabulary Check (page A56)1. equation 2. solve 3. identities; conditional4. 5. extraneous6. quadratic equation7. factoring; extracting square roots; completing the

square; Quadratic Formula

1. Identity 2. Conditional equation3. Conditional equation 4. Identity 5. Identity6. Identity 7. Identity 8. Identity9. Conditional equation 10. Conditional equation

11. 4 12. 13. 14. 3 15. 516. 17. 9 18. 19. No solution20. All real numbers 21. 22.23. 24. 6 25. 9 26. 5027. No solution. The x-terms sum to zero.28. No solution. The x-terms sum to zero. 29. 1030. 31. 4 32. 0 33. 3 34. 35. 0

36. 37. No solution. The variable is divided out.

38. 39. No solution. The solution is extraneous.

40. 41. 2 42.43. No solution. The solution is extraneous.44. No solution. The solution is extraneous.

45. 0 46. 47. All real numbers 48. No solution. The x-terms sum to zero.49. 50.51. 52.53. 54.55. 56. 57. 58. 9, 159. 60. 61. 62.

63. 64. 2, 6 65. 66.67. 68. 69. 70.71. 72. 73. 74.75. 76. 77.

78. 79. 80.

81. 2 82. 83. 84.

85. 86.

87. 88. 89.

90. No solution 91. 92.

93. 94. 95. 96.97. 98. 99.

100. 101. 102.

103. 104. 105.

106. 107. 108.

109. 110. 111.

112. 113. 114.

115. 116.

117. 118.119. 120. 2.137, 18.063121. 122.123. 124.125. 126. 127. 128. 7

129. 130. 131.

132. 133. 134.

135. 136. 137. 138.

139. 140. 141. 142.143. 144. 145. 146.147. 148. 149.

150. 151. 152. 153. 50

154. 155. 26 156. 157.

158. 159. 160. 161. 0162. No solution 163. 9 164. 1 165.

166. 167. 168. 169. 1

170. 171. 172.

173. 174. 175. 4, �5�3, 1�3 ± �21

6

2, �122, �320, 1, 35

1 ± 5�52

±�14�29, 25

�3 ± 16�23, �22, �5124

3

�16�4916

� 3�21, �2±�7

6

±12, ±4±2±�3, ±1

±2±1�23, 1, �10, 43�3, 08

3�6

±2±30, ±5

20, ±

3�2

2

±134

±�97

4�

b

a,

b

a

�12�

32 ± �31

2 ± �3

6, �120, �31 ± �2�2.995, 2.971�0.290, �2.200

0.672, �0.9681.687, �0.4881.355, �14.071

1.400, �0.1500.976, �0.643

686 ± 196�625

�38

±�265

8

�7 ± �136 ± �11�8

5±�3

5

2 ±�6

2�

32

±�13

227

5

4±�5

2�

1

2± �2�

1

3±�11

6

�43

5

4±�3

4

2

3±�7

3

1

2±�5

2�4 ± 2�5�3 ± �13

�7 ± �55 ± �31 ± �3

35, 15

14, �3

41, �12

12, �1

112

, �92

�5 ± �894

2 ± 2�32

3± �21 ±

�63

�4 ± �2��11 � 6�11 � 6,

3, �14, �8�92

�74

±�11

21 ± 3�2

25 ± �30

�2 ± �14�18, �88, 16±2±3�3±4�2±�11

±13±7�a � b, �a � b�a�8, 16�

203 , �42, �6

�32, 113, �1

2�32�5

4, �2�13, 130, �1

2

4x2 � 2x � 1 � 03x2 � 90x � 10 � 0�3x2 � 42x � 134 � 0x2 � 6x � 6 � 0x2 � 16x � 02x2 � 8x � 3 � 0

x�12

74�

133

116

53

97

12

�65

�5�4x�

58�5

�9�12

ax � b � 0

x 1.

x 1.x �1, 1.

Year 2002 2003 2004

Ratio 0.6333 0.6592 0.6861

Year 2005 2006 2007

Ratio 0.7013 0.7003 0.6812



176. 177. 178. 179.

180. 181. 182.

183. 184.

185. (a) 61.2 inches(b) Yes. The estimated height of a male with a 19-inch

femur is 69.4 inches.(c)

100 inches(d) There would not be a problem because it

is not likely for either a male or a female to be 100inches tall (which is 8 feet 4 inches tall).

186. 23,437.5 miles187. after about 28 hours188. (a) (b)

(c)

189.

190.

191.

192. miles per hour and 600 miles per hour193. (a) 1998 (b) During 2007194. 26,250 passengers 195. 500 units196. 250,000 units197.

The equation cannot be written in the form 198. False. The product must equal zero for the Zero-Factor

Property to be used.199. False. See Example 14 on page A55.200. False. has only one solution to check, 0.201. Equivalent equations have the same solution set, and one

is derived from the other by steps for generating equiva-lent equations.

202. Remove symbols of grouping, combine like terms, reducefractions.Add (or subtract) the same quantity to (from) each side ofthe equation.Multiply (or divide) each side of the equation by the samenonzero quantity.Interchange the two sides of the equation.

203. Yes. The student should have subtracted 15x from bothsides to make the right side of the equation equal to zero.Factoring out an x shows that there are two solutions,

and 204. (a) and (b)

(c) The method used in part (a) reduces the number ofalgebraic steps.

205. 206.207. 208.209. 210.211.212. Isolate the absolute value by subtracting x from both sides

of the equation. The expression inside the absolute valuesigns can be positive or negative, so two separate equa-tions must be solved.

213. (a) (b)


Vocabulary Check (page A66)1. solution set 2. graph 3. negative4. solution set 5. double 6. union

1. Bounded 2. Bounded3. Unbounded 4. Unbounded5. Unbounded 6. Unbounded7. b 8. f 9. d 10. c 11. e 12. a

13. (a) Yes (b) No (c) Yes (d) No14. (a) No (b) No (c) Yes (d) No15. (a) Yes (b) No (c) No (d) Yes16. (a) No (b) No (c) Yes (d) No17. (a) Yes (b) Yes (c) Yes (d) No18. (a) No (b) Yes (c) No (d) Yes19. 20.

21. 22.

23. 24.

3 4 5 6 7

x

10 11 12 13 14

x

x ≤ 5x ≥ 12

2− 5

−1−2−3−4

x

2 310−1

x

−2

32

x < �52x < 3

2

−2−6 −5 −4 −3

x

1 2 3 4 5

x

x < �4x < 3

x ≤ 7.x < �2.x ≥ �5.x > 11.

2 < x ≤ 10.�1 ≤ x ≤ 5.

x � 0, 1x � 0, �ba

a � 9, b � 9x2 � 6x � 4 � 0x2 � 2x � 1 � 0

30x2 � 7x � 2 � 0x2 � 22x � 112 � 0x2 � 15x � 44 � 0x2 � 3x � 18 � 0

x � �5, �103

x � 6.x � 0

2x � 5, 2x � 3 � 8

�x� � 0

ax � b � 0. 3x � x2 � 10

False. x�3 � x� � 10

�550

20�33

� 11.55 inches

5�2

2� 3.54 centimeters

6 inches � 6 inches � 2 inches

l � 48 feet w � 34 feetw�w � 14� � 1632

w

w + 14

y � �0.25t � 8;

x � 100.59;

10, �13, �1 � �17

2

�6, �3, 3�3, �353, �3

3, �2±11 ± �31

3

3

4, �1

Height, Female Malex femur length femur 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



AP

PE

ND

IX A

25. 26.

27. 28.

29. 30.

31. 32.

33. 34.

35. 36.

37. 38.

39. 40.

41. 42.

43. 44. No solution

45. 46.

47. 48.

49. No solution 50. No solution51. 52. All real numbers x

53. 54.

55. 56.

57. 58.

59. 60.

61. 62.

63. 64.

65. 66.

67. 68.

69. 70.

(a) (a)

(b) (b)71. 72.

(a) (a)(b) (b) x ≥ 8

3x ≤ 4

53 ≤ x ≤ 3�2 ≤ x ≤ 4

−3

9−9

9

−6 6

−2

6

x ≥ �32x ≤ 3

2

x ≤ 6x ≥ 2

−5 7

−2

6

−4 8

−5

3

�7 ≤ x ≤ 5x ≤ �272 , x ≥ �

12

−10 10

−10

10

−15 1

−10

10

x < �11, x > 2�6 ≤ x ≤ 22

−15 9

−10

1010

−10

−10 24

x < 2x ≤ 2

−10 10

−10

10

−10 10

−10

10

x ≤ 2x > 2

−10 10

−10

10

−10 10

−10

10

2

755

1

1

0

x

−12

− −

−16 −4−8

x

112

292

15 ≤ x ≤ 7

5x ≤ �292 , x ≥ �

112

−35 −28 −21 −14 70−7

xx

3 654

x < �28, x > 04 < x < 5

−1 0 1 2 3 4

x

0−5−10−15 5 10 15

x

11

0 < x < 3x ≤ �5, x ≥ 11

−1−2 10 2 3

x

−2 −1 10 2 3 4

x

32

−

�2 < x < 3x ≤ �32, x ≥ 3

1 2 30−3 −2 −1

x

201510 25 30

x

14 26

14 ≤ x ≤ 26

−20 −10 0 10

x

20−2−3 −1 0 1 2 3

x

x < �15, x > 15x < �2, x > 2

−6 −4 −2 0 2 4 6

x

−2−4−6 0 2 4 6

x

x < �4, x > 4�6 < x < 6

121110 13 14

x

13.510.5

10.5 ≤ x ≤ 13.5

1 3 5 7 9 11

x

0 1−1

−

x

−43

41

3 < x < 9�34 < x < �

14

−2−4

−3 7

86420

x

−2−4−6

−

0 2 64 8

x29

215

�3 ≤ x < 7�92 < x < 15

2

10−1−2−3−4−5−6

x

30 1 2−1

x

�6 < x ≤ 1�1 < x < 3

15 16 17 18

x

14−2−3−4−5−6

x

x > 16x ≥ �4

0 21 3

x

−1

16

210 3 4

x

x > 16x ≥ 2

5 6 7 8 9

x

432 5 6

x

x < 7x ≥ 4

−2

2

−1

−1

0 1x

43 5 6 7

x

x < �12x < 5

x

−1−2−3−4−5

−2 −1 10 2

x

27

x ≥ � 3x ≥ 27

−1 0 1

1

2

2x

0 1 2 3 4

x

x ≥ 12x > 2



73. 74.

(a) (a)

(b) (b)

75. 76. 77.

78. 79. 80.81. All real numbers within eight units of 1082. All real numbers more than four units from 883. 84. 85.86. 87.88. 89. 90.91. 92. More than 42,857 copies93. 94. 95.96. 97. 98. 24 weeks99. (a) (b)

100. (a) and (b)

(c) (d)(e) An athlete’s weight is not a particularly good indica-

tor of the athlete’s maximum bench-press weight.Other factors, such as muscle tone and exercise habits,influence maximum bench-press weight.

101. (a) (b)102. (a) (b)103.104.

centimeters105. Might be undercharged or overcharged by $0.19.106. $0.47107. 108.

109.110. (a)

(b)(c)(d)

111. False. c has to be greater than zero.112. False. If then and 113. b114. One set (the solution is not unique):


Vocabulary Check (page A75)1. numerator 2. reciprocal

1. Change all signs when distributing the minus sign.

2. The 3 is distributed to both terms.

3. Change all signs when distributing the minus sign.

4. The expression on the right should be negated.

5. z occurs twice as a factor.

6. yz is one term, not two. 7. The fraction as a whole is multiplied by a, not the

numerator and denominator separately.

8. The exponent also applies to the coefficient. 9. cannot be simplified.

10. Do not apply radicals term-by-term.

11. Divide out common factors, not common terms.

cannot be simplified.

12. cannot be simplified.

13. To get rid of negative exponents:

14. The negative exponent is on a term of the denominator,not a factor.

15. Factor within grouping symbols before applying exponentto each factor.

16. Exponents are applied before multiplying.x�2x � 1�2 � x�4x2 � 4x � 1� � 4x3 � 4x2 � x

�x2 � 5x�1�2 � �x�x � 5��1�2 � x1�2�x � 5�1�2

1x � y�1 �

yxy � 1

, y 0

1a�1 � b�1 �

1a�1 � b�1 �

abab

�ab

b � a.

6x � y6x � y

2x2 � 15x

�25 � x2 � ��5 � x��5 � x�

�x � 9�4x�2 � 16x2

axy �

axy

x�yz� � xyz�5z��6z� � 30z2

1 � x�5 � x��x� � �

x � 1x�x � 5�

416x � �2x � 1� �

414x � 1

5z � 3�x � 2� � 5z � 3x � 6

2x � �3y � 4� � 2x � 3y � 4

a � 1, b � 5, c � 5

�x ≥ �8.10 ≥ �x�10 ≤ x ≤ 8,

v < 500 vibrations�second1.2 millimeters ≤ t ≤ 2.4 millimeters� 3.6 millimeters� 330 vibrations�second

20 ≤ h ≤ 80

6968676665 70 71 72h

71.265.8

1615141312 17 18 19

t

17.513.7

65.8 ≤ h ≤ 71.213.7 < t < 17.5

573.6025 square centimeters ≤ area ≤ 597.8025 square106.864 square inches ≤ area ≤ 109.464 square inches

t ≥ 172 ≤ t ≤ 7t > 161 ≤ t ≤ 10

x ≥ 181.54x ≥ 182

140140 260

300

x ≥ 129

075 150

5

134 ≤ x ≤ 234x ≥ 16,394x ≥ 36r > 5%r > 3.125%

x > 6�x � 6� ≤ 7�x � 3� > 4�x � 8� ≥ 5

�x � 12� < 10�x � 1� ≤ 4�x � 7� ≥ 3�x� > 3�x� ≤ 3

��52, ��, 72��, 3�

��3, ��10, ��5, ��x ≤ �4, x ≥ 0x ≤ �1, x ≥ 7

�10 ≤ x ≤ 61 ≤ x ≤ 5

−5 1

−1

3

−5 10

−2

8



AP

PE

ND

IX A

17. To add fractions, first find a common denominator.

18. Be careful when using a slash to denote division.

19. 20. 21.22. 23. 24. 25. 2 26.

27. 28. 29. 30.

31. 1, 2 32. 4, 7 33. 34.35. 36. 37.38. 39.40.

41.

42. 43.

44. 45.

46. 47.

48. 49. 50.

51. 52.

53. 54.

55. 56.

57. 58.

59.

60.

61. (a)

(b)

(c)

62. (a) Answers will vary.(b)

63.

64. False. Cannot move term-by-term from denominator tonumerator.

65.

66. False. does not factor into 67. Add exponents when multiplying powers with like bases.

68. There is no error.69. When a binomial is squared, there is also a middle term.

70. There is no error.71. The two answers are equivalent and can be obtained by

factoring.

(a) (b) 815�4 � x�3�2�x � 1�2

5�2x � 3�3�2�x � 1� � 1

15�2x � 1�3�2�3x � 1� � 4

60�2x � 1�3�2�3x � 1� � 1

60�2x � 1�3�2�12x � 4� � 1

60�2x � 1�3�2�6�2x � 1� � 10�

110�2x � 1�5�2 �

16�2x � 1�3�2

�xn � yn�2 � x2n � 2xnyn � y2n x2n � y2n

xn � x3n � x4n

��x � 3��x � 3�.x2 � 9

True. 1�x � 4

�1

�x � 4��x � 4�x � 4

��x � 4x � 16

True. x�1 � y�2 �1x

�1y2 �

y2 � xxy2

3x�x2 � 8x � 20 � �x � 4��x2 � 46�x2 � 4�x2 � 8x � 20

x � 0.5 mile

3�x � 6�1�2�6x � 4 � �x � 6�5�2�2�3x � 2�3�2

�3x � 2�1�2�15x2 � 4x � 45�2�x2 � 5�1�2

2�3x2 � 5x � 6��x2 � 6��2x � 5�

xx2 � 4

1 � 4x

�x � 1�2�2x � 3x 2�1�2

4x � 3

�3x � 1�4�3

x � 3

�2x � 1�1�2

�1

�x � 3�2�3�x � 2�7�4

�6�2x � 3��4x2 � 9�3�2

27x2 � 24x � 2

�6x � 1�4

�11x2 � 5

x6�x2 � 1�4

�7x2 � 4x � 9

�x2 � 3�3�x � 1�4

x

3�

5x2

3

3

x1�2� 5x3�2 � x7�22x7�2 � 3x3�2 �

5

x1�2�

1

x3�2

4x8�3 � 7x5�3 �1

x1�3x � 5 �

4

x2

16

x� 5 � xx�x � 2��1 � x�2 �

83

�9x��3

43

x�1 � 4x�4 � 7x�2x��1�3

�x � 1�x�1�6 � x��1�2

3x2�2x � 1��34t � 33x � 11 � 10x2 � 20x31 � 7x

10x � 31 � 5x

4

3,

16

9259

, 4916

x � 11

2x 2

12�

14

133x � 2

2x2 � x � 15x23x � 2

�1�2�y �12

� y �y2

3x

�4y

�3y � 4x

xy

x 0.5 1.0 1.5 2.0

t 1.70 1.72 1.78 1.89

x 2.5 3.0 3.5 4.0

t 2.02 2.18 2.36 2.57

x 0 1 2

0 2.9 8.7 12.5

0 2.9 8.7 12.5�1.1�2.9�8.7y2

�1.1�2.9�8.7y1

52�

12�1�2


appendix a x appendix a.1 (page a8) -...

Documents