4-1 inequalities and their graphs - weebly

Inequalities and Their Graphs

200 Chapter 4 Solving Inequalities

Lesson 1-3

Graph the numbers on the same number line.

1. 4 2. 23 3. 4. 0 5. 1.5

Complete each statement with <, 5, or >.

6. 23 j 25 7. 4.29 j 4.8 8. (23)(24) j 12

9. 21 2 2 j 6 2 9 10. j 11. 1 j 1

New Vocabulary • solution of an inequality

12

12

13

132

452

34

93

What You’ll Learn• To identify solutions of

inequalities

• To graph and writeinequalities

. . . And WhyTo write inequalities forspeed limits and startingsalaries, as in Example 5

Part 1

A is any number that makes the inequality true. Forexample, the solutions of the inequality x , 3 are all numbers that are less than 3.

Identifying Solutions by Mental Math

Is each number a solution of x # 7?

a. 9 No, 9 # 7 is not true.

b. 21 Yes, 21 # 7 is true.

c. 5 7; yes, # 7 is true.

Is each number a solution of x $ 24.1?a. 25 no b. 24.1 yes c. 8 yes d. 0 yes

You can determine whether a value is a solution of an inequality by evaluating an expression.

Identifying Solutions by Evaluating

Is each number a solution of 2 2 5x . 13?

a. 3 b. 24

2 2 5x . 13 2 2 5x . 13

2 2 5(3) . 13 d Substitute for x.S 2 2 5(24) . 13

2 2 15 . 13 d Simplify.S 2 1 20 . 13

213 � 13 d Compare.S 22 . 13

3 does not make the original 24 does make the original inequality true, so 3 is not a solution. inequality true, so 24 is a solution.

EXAMPLEEXAMPLE22

11Quick Check

142

142

142

EXAMPLEEXAMPLE11

solution of an inequality

4-14-1

11 Identifying Solutions of Inequalities

1–5.

S

≠

≠R

RS

Check Skills You’ll Need GO for Help

�2�1�4�3 0 2 41 3

Less than and is less thanhave different meanings.For example, x less than 3means 3 - x; x is less than3 means x , 3.

Vocabulary Tip

4-14-1

200

1. PlanObjectives1 To identify solutions of

inequalities2 To graph and write

inequalities

Examples1 Identifying Solutions

by Mental Math 2 Identifying Solutions

by Evaluating3 Graphing Inequalities4 Writing an Inequality From

a Graph5 Real-World Problem Solving

Math Background

Graphing solutions on a numberline helps students understandthat inequalities have more thanone solution.

More Math Background: p. 198C

Lesson Planning andResources

See p. 198E for a list of theresources that support this lesson.

Bell Ringer Practice

Check Skills You’ll NeedFor intervention, direct students to:

Exploring Real Numbers Lesson 1-3: Example 4Extra Skills and Word

Problems Practice, Ch. 1

PowerPoint

Special NeedsHave students model inequalities using a set of scaleswith blocks of different weights. Remind studentsthat an equation means that the scales are in balance,while an inequality means that the scales are out of balance.

Below LevelHelp students to understand that expressions such as -1 � a and a � -1 have the same meaning.Review several examples of these expressions with the students.

L2L1

learning style: tactile learning style: verbal

Lesson 4-1 Inequalities and Their Graphs 201

Is each number a solution of 6x 2 3 . 10?a. 1 no b. 2 no c. 3 yes d. 4 yes

Part 2 Graphing and Writing Inequalities in One Variable

You can use a graph to indicate all of the solutions of an inequality.

Inequality Graph

x , 3 The open dot shows that 3 is not asolution. Shade to the left of 3.

m $ 22 The closed dot shows that –2 is asolution. Shade to the right of –2.

21 $ a The closed dot shows that –1 is asolution. Shade to the left of –1.

You can also write 21 $ a as a # 21.

Graphing Inequalities

a. Graph c . 22.

The solutions of c S –2 are all the points to the rightof 22.

b. Graph 4 # m.

The solutions of 4 K m are 4 and all the points to theright of 4.

Graph each inequality.a. a , 1 b. n $ 23 c. 2 . p

You can write an inequality for a graph.

Writing an Inequality From a Graph

Write an inequality for each graph.

a. x , 24 Numbers less than –4 are graphed.

b. x # 5 Numbers less than or equalto 5 are graphed.

c. x . Numbers greater than are graphed.

d. x $ 21 Numbers greater than or equalto –1 are graphed.

Write an inequality for each graph. Choice of variable may vary.

a. x L 2 b. x R 0�3 �2 �1 0 1 2 3�3 �2 �1 0 1 2 3

44Quick Check

�3 �2 �1 0 1 2 3

12

12�3 �2 �1 0 1 2 3

�1 0 1 2 3 4 5

�5 0 1�4 �3 �2 �1

EXAMPLEEXAMPLE44

33Quick Check

�1 0 1 2 3 4 5

�3 �2 �1 0 1 2 3

EXAMPLEEXAMPLE33

�3 �2 �1 0 1 2 3

�3 �2 �1 0 1 2 3

�1 0 1 2 3 4 5

22Quick Check

12 Graphing and Writing Inequalities in One Variable

�2�1 0 1 2 �3�2�5�4 �1 10 �1 0 1 2 53 4

You normally read 21 $ a as “21 isgreater than or equal toa.” The inequality symbolalso indicates that a is lessthan or equal to 21.

Vocabulary Tip

nline

Visit: PhSchool.comWeb Code: ate-0775

201

2. Teach

Guided Instruction

Math Tip

The inequality sign replaces theequal sign in a math sentence. The equal sign is the mathematicalway of writing is. Therefore, theinequality sign is the mathematicalway of writing is less than.

Additional Examples

Is each number a solution of x $ 5?a. -2 nob. 10 yesc. yes

Is each number a solution of 3 + 2x � 8? a. -2 yesb. 3 no

Error Prevention

Tell students: Write an inequalityfor Kyle is older than Jaime.Kyle’s age S Jaime’s age Write this using a less than sign.Jaime’s age R Kyle’s age Leadstudents to understand the twoinequalities have the samemeaning.

EXAMPLEEXAMPLE33

22

255

11

EXAMPLEEXAMPLE11

Advanced LearnersAsk students to discuss other situations that could be represented by an inequality and determinereasonable solutions.

English Language Learners ELLSome students may be confused by the terminequality. In the word inequality, underline in. Havestudents discuss the meanings of inactive, incorrect,and informal. Ask: What syllable do all these wordshave in common? in, meaning not

L4

learning style: verballearning style: verbal

PowerPoint

202

Careers

Many salespeople are paid asalary plus commission. Askstudents to write an inequalityshowing what a salesperson witha salary of $700 per week pluscommission gets paid. Sampleanswer: p L 700

Additional Examples

a. Graph d � 3.

b. Graph -3 $ g.

Write an inequality for each graph.a. x R 2

b. x K–3

c. x S–2

d. x S

Define a variable and write aninequality for each situation.a. A speed that violates the lawwhen the speed limit is 55 mphLet v ≠ an illegal speed; v S 55b. A job that pays at least $500 a month Let p ≠ pay per month; p L 500

Resources• Daily Notetaking Guide 4-1• Daily Notetaking Guide 4-1—

Adapted Instruction

Closure

Have students explain thedifference between the open dotand the closed dot in graphinginequalities. An open dot showsthat a number is not a solution; aclosed dot shows that a number is a solution. For example, for x R 3 or x S 3, the dot at 3 wouldbe open. For x K 3 or x L 3, thedot at 3 would be closed.

L1

L3

55

12�3�2�1 0 1 2 3

�3�2�1 0 1 2 3

�3�4 �2�1 0 1 2

�1 0 1 2 3 4 5

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�3�4 �2�1 0 1 2

�1 0 1 2 3 4 5

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EXAMPLEEXAMPLE55

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20.�4�5 �3 �2 �1 0

�1 0 1 3221.

22.

23.�4�5 �3�2�1 0

�2�1 0 21

�6�7 �5�4�3�224.

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PowerPoint


You can describe real-world situations using an inequality.

Define a variable and write an inequality for each situation.

a. b.

Let s 5 a legal speed. Let p 5 pay per hour (in dollars).The sign indicates that s # 65. The sign indicates that p $ 6.15.

a. Critical Thinking In part (a) of Example 5, can the speed be all real numbersless than or equal to 65? Explain. See left above.

b. In part (b) of Example 5, are all real numbers greater than or equal to $6.15reasonable solutions of the inequality? Explain. See left above.

EXERCISES For more practice, see Extra Practice.

Practice and Problem Solving

Mental Math Is each number following the inequality a solution of the given inequality?

1. v $ 25; 4 yes 2. 0.5 . c ; 2 no 3. b , 4; 20.5 yes 4. d # ; 5 yes

5. g # ; 3 no 6. k , 0; 21 yes 7. a . 3.2; 3 no 8. x $ 22.5; 22.5

Is each number a solution of the given inequality?

9. 3x 2 7 . 21 a. 2 no b. 0 no c. 5 yes

10. 4n 2 3 # 5 a. 2 yes b. 3 no c. 21 yes

11. 2y 1 1 , 23 a. 0 no b. 22 no c. 1 no

12. $ 5 a. 0.5 yes b. 2 no c. 24 no

13. n(n 2 3) , 54 a. 9 no b. 3 yes c. 10 no

14. 5(2q 2 8) $ 7 a. 22 no b. no c. 6 yes

Match each inequality with its graph.

15. x , 4 C 16. x $ 4 B 17. x . 4 D 18. x # 4 A

A. B.

C. D.�2 �1 0 1 2 3 4�3 5�2 �1 0 1 2 3 4�3 5

�2 �1 0 1 2 3 4�3 5�2 �1 0 1 2 3 4�3 5

Example 3(page 201)

92

4 2 mm

Example 2(page 200)

125

173

Example 1(page 200)

55Quick Check

EXAMPLEEXAMPLE Real-World Problem Solving55

Practice and Problem SolvingFor more exercises, see Extra Skill and Word Problem Practice.EXERCISES

Practice by ExampleAA

5a. No; speeds cannot be negative, so youcan’t use all realnumbers.

b. No, answers mayvary. Sample:Hourly wages arenot likely to be inhundreds ofdollars.

yesGO for

Help

Graph each inequality. 19–26. See margin p. 202.

19. x . 1 20. s , 23 21. y # 24 22. t $ 21

23. 22 , d 24. # b 25. 7 $ a 26. 4.25 . c

Write an inequality for each graph. Choice of variable may vary.

27. 28.

29. 30.

31. 32.

Define a variable and write an inequality to model each situation.

33. A bus can seat at most 48 students. Let s ≠ number of students. s K 48

34. In many states, you must be at least 16 years old to obtain a driver’s license.

35. It is not safe to use a light bulb of more than 60 watts in this light fixture.

36. At least 350 students attended the band concert Friday night.

37. Aviation The Navy’s flying squad, the Blue Angels, makes more than 75appearances each year. Let a ≠ number of appearances. a S 75

Write each inequality in words. 38–46. See left. 47–49. See margin.

38. n , 5 39. b . 0 40. 7 $ x 41. z $ 25.6

42. 4 . q 43. 21 $ m 44. 35 $ w 45. g 2 2 , 7

46. a # 3 47. 6 1 r . 22 48. 8 # h 49. 1.2 . k

50. Writing Explain how you choose whether to draw an open or a closed dotwhen you graph an inequality.

51. Open-Ended Describe a situation that you can represent using the inequality x $ 18. Answers may vary. Sample: Every class has at least 18 students.

52. Multiple Choice Suppose your school plans a musical. The director’s goal isticket sales of at least $4000. Adult tickets are $5.00 and student tickets are$4.00. Let a represent the number of adult tickets and s represent the numberof student tickets. Which inequality represents the director’s goal? D

Rewrite each inequality so that the variable is on the left. Then graph the solutions.

53. 2 , x x S 2 54. 25 $ b b K –5 55. 0 # r r L 0 56. 5 . a a R 5

Graph each inequality from the given description.

57. t is nonnegative. 58. x is positive.

59. k is no more than 3. See margin. 60. r is at least 2.

61. s is at most 4. See margin. 62. v is no less than 7. See margin.

63. Writing Explain how you interpret the phrases “at least” and “at most” in aninequality that models a real-world situation.

�2�1 0 1 2�2�1 0 1 2

5a 1 4s $ 40005a 1 4s # 40005a 1 4s . 40004a 1 5s , 4000

Apply Your SkillsBB

Example 5(page 202)

�3 �2 �1 0 1 2 3�4 42 3 4 5�2 610�1

�6 �5 �4 0�7 1�3 �2 �1�3 �2 �1 0 1 2 3�4 4

3 4 5 6�1 7210�3 �2 �1 0 1 2 3�4 4

Example 4(page 201)

232

Let a ≠ age. a L 16

Let s ≠ number of students. s L 350

Let w ≠ acceptable number of watts. w K 60

38. n is less than 5.

39. b is greater than 0.

40. 7 is greater than orequal to x, or x is lessthan or equal to 7.

41. z is greater than orequal to –5.6.

42. q is less than 4, or 4 isgreater than q.

43. –1 is greater than orequal to m, or m is lessthan or equal to –1.

44. 35 is greater than orequal to w, or w is lessthan or equal to 35.

45. g minus 2 is less than 7.

46. a is less than or equalto 3.

Use an open dot for R or S. Use a closeddot for K or L.

“At least” is translated as L.“At most” is translated as K.

53–56. See margin for graphs.

x S –3 x K 7

x L 1

x L 4.5

x R –6

x R –0.5

10 2 3 4 5


For help withcounterexamples see p. 18.

for HelpGO

53.

54.

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�4�3�6�5 �2 0�1

�1 0 1 2 347. 6 plus r is greater than –2.

48. 8 is less than or equal to h,or h is greater than or equalto 8.

49. 1.2 is greater than k, or k isless than 1.2.

56.

59.10 2 3 4 5

2 3 4 5 6 61.

62.�2 0 2 4 6 8

�1 0 1 2 53 4

3. PracticeAssignment Guide

A B 1-14, 51, 66-68

A B 15-50, 52-65C Challenge 69-73

Test Prep 74-79Mixed Review 80-94

Homework Quick CheckTo check students’ understandingof key skills and concepts, go overExercises 10, 31, 40, 63, 65.

Error Prevention!

Exercises 33–37 Some studentsmay write � for at most and �for at least. Make these termsrelevant to the students’ lives.Ask: If 3 hours is at most theamount of time you want tostudy tonight, do you want tostudy more than 3 hours? no Ifyou want to get at least $10 foran allowance, do you want lessthan $10? no

2

1

Guided Problem SolvingGPS

Enrichment

Reteaching

Adapted Practice

Practice 4-1 Ratio and Proportion

Find each unit rate.

1. $60 for 8 h 2. 3. 4.

5. An 8-ounce bottle of lotion costs $4.50. What is the cost per ounce?

6. A pound of coffee costs $14.99. What is the cost per ounce?

Which pairs of ratios could form a proportion? Justify your answer.

7. 8. 9. 10. 11.

Solve each proportion.

12. 13. 14. 15.

16. 17. 18. 19.

20. 21. 22. 23.

24. 25. 26. 27.

28. 29. 30. 31.

32. 33. 34. 35.

36. 37.

38. 39.

40. 41.

42. You are riding your bicycle. It takes you 28 min to go 8 mi. If youcontinue traveling at the same rate, how long will it take you to go 15 mi?

43. Suppose you traveled 84 mi in 1.5 h. Moving at the same speed, howmany mi would you cover in 3 h?

44. A canary’s heart beats 130 times in 12 s. Use a proportion to find howmany times its heart beats in 50 s.

45. Your car averages 18 mi per gal on the highway. If gas costs $1.85 per gal, how much does it cost in dollars per mi to drive your car on the highway?

14

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Use the map below for Exercises 64–65.

64. Air Travel You plan to go from New York City to Los Angeles. Let x be thedistance in miles of any air-route between New York City and Los Angeles. Theshortest route is a direct flight. Using the map, write a true statement about themileage of any route from New York City to Los Angeles. x L 2451

65. Air Travel Your travel agent is making plans for you to go from Chicago toNew Orleans. A direct flight costs too much. Option A consists of flights fromChicago to Dallas to New Orleans. Option B consists of flights from Chicago toOrlando to New Orleans. Write an inequality comparing the mileage of thesetwo options. Option A R Option B

66. Error Analysis A student claims that the inequality 3x 1 1 . 0 is always truebecause multiplying a number by three and then adding one to it makes thenumber greater than zero. Use a counterexample to show why the student isnot correct.

67. Critical Thinking Describe how you can display the solutions of the inequalityx 3 on a number line.

68. Critical Thinking Explain the difference between “4 greater than x” and“4 . x.”

69. Critical Thinking Which is the correct graph of 24 , 2x? Explain.

A. B.

C. D.

70. Reasoning Give a counterexample for this statement. If a , b, then a2 , b2.

71. Reasoning Describe the numbers a and b for which the followingstatement is true. If a , b, then a2 5 b2.

Graph on a number line.72. all values of x such that x . 22 and x # 2

73. all values of x such that x , 21 or x . 3 �2 �1 0 1 2 3 4 5

�2�3 �1 0 1 2 3

�4 �3 �2 2�5 3�1 0 1�2 �1 0 1 2 3 4�3 5

�4 �3 �2 2�5 3�1 0 1�2 �1 0 1 2 3 4�3 5

ChallengeCC

2

713 mi

1745 mi

1673 mi

798

mi

1251 mi

939

mi

985m

i

821

mi

2212 mi

433 mi

962 mi

1374 mi

1158

mi

528 mi

2451 mi

OrlandoNew Orleans

LosAngeles

ChicagoNew YorkCity

Dallas

U N I T E D S T A T E S

C A N A D A

M E X I C O

N

S

E

W

Airline route

0 150 300 mi

0 150 300 km

ConnectionReal-World

There is an average of 2500flights in and out ofChicago’s O’Hare Airporteach day.

“4 greater than x” means x ± 4; “4 S x” means 4 is greater than x.

C; the inequality is true for x = 3 but not true for x = 5, so C is correct.

Answers may vary. Sample: a ≠ –1, b ≠ 12

a is negative, and a and b are opposites.

Problem Solving Hint

In Exercise 71, makelogical choices forvalues of a and b. Usepositive and negativenumbers as well as zero.

Answers may vary. Sample: For x ≠ –1, 3(–1) ± 1 ≠–3 ± 1 ≠ –2. –2 � 0.

Put an open dot at 3 and color the rest of thenumber line.

GPSGO nline

Homework HelpVisit: PHSchool.comWeb Code: ate-0401

204

Lesson Quiz

1. Is each number a solution of x� -1?a. 3 yesb. -5 no

2. Is 2 a solution of 3x - 4 � 2? no

3. Graph x � 4.

4. Write an inequality for thegraph. p K –2

5. Graph each inequality.a. t is at most 2.

b. w is at least 1.

Alternative Assessment

Make a number line on the floorout of masking tape. Have astudent write an inequality on theboard. Have another studentdemonstrate the inequality onthe number line by standing onthe appropriate number whileholding their arms close to theirsides or in a circle and thenwalking down the number line inthe correct direction. If “or equalto” is part of the inequality, havestudents tuck their arms in closeto their side. If there is no “orequal to,” have students make abig circle with their arms. Forexample, to demonstrate x � 2,the student stands on 2, makes an arm circle, and walks in thepositive direction.

�2�1 0 1 2 3 4

�3�2�1 0 1 2 3

�3�2�1 0 1 2 3

�1 0 1 2 3 4 5

PowerPoint

4. Assess & Reteach


Standardized Test Prep

74. Which inequality has the same solutions as n . 5? CA. n , 25 B. n , 5 C. 5 , n D. 2n . 25

75. What is the least whole-number solution of k $ 25? HF. 25 G. 24 H. 0 J. 1

76. Employees must work at least 20 years in a company in order to receive fullbenefits upon retirement. Which inequality or graph does NOT describethis situation? BA. y $ 20 B. y . 20C. 20 # y D.

77. Which value makes the inequality x2 $ x false? H

F. G. 0 H. J. 1

78. Fire codes require that no more than 150 persons occupy a conferenceroom. Which graph includes a room count in possible violation of the fire codes? B

A.

B.

C.

D.

79. A stretch of mi of highway is being repaired. The project foremanreported that less than half of the job is complete. Draw a diagram to showthe remaining miles to be repaired. Then write an inequality for thenumber of miles m that still need repair. See above.

Mixed Review

A right triangle has hypotenuse c and legs a and b. Find the missing side. Ifnecessary, round to the nearest tenth.

80. a 5 6, b 5 12 13.4 81. b 5 5, c 5 8 6.2 82. a 5 1.2, b 5 0.5 1.3

83. c 5 16, b 5 7 14.4 84. a 5 10, c 5 30 28.3 85. a 5 , b 5 1.4

Determine whether the given lengths can be sides of a right triangle.

86. 10, 13, 27 no 87. 9, 40, 41 yes 88. 28, 96, 100 yes

Name the property that each equation demonstrates.

89. 3 (2 ? 7) 5 (3 ? 2) 7 90. 5 3 1 5 1 3 5 91. 3 1 4 5 4 1 3

Find each measure for the following data.

1 3 4 4 5 5 7 9 9 9 13

92. mean 6. 93. median 5 94. mode 927

Lesson 1-6

Lesson 2-5

87

67

Lesson 3-9

12 35Short Response

148147 151149 150

148147 151149 150

148147 151149 150

148147 151149 150

1421

4

1050 2515 20

Multiple Choice

Test Prep

Mixed ReviewMixed Review

79. [2]

Since 6 is half, m S 6 .

[1] incorrect inequality OR no diagram

310

310

to berepaired

completed

12 35

CommutativeProperty of Addition

Commutative Propertyof Multiplication

Associative Propertyof Multiplication

GO forHelp

lesson quiz, PHSchool.com, Web Code: ata-0401

205

Test Prep

ResourcesFor additional practice with avariety of test item formats:• Standardized Test Prep, p. 247• Test-Taking Strategies, p. 242• Test-Taking Strategies with

Transparencies

Solving Inequalities UsingAddition and Subtraction

Lesson 2-1 and Review page 118

Complete each statement with R, ≠, or S.

1. 23 1 4 ■ 25 1 4 2. 23 1 6 ■ 4 1 6 3. 23.4 1 2 ■ 23.45 1 2

Solve each equation.

4. x 2 4 5 5 9 5. n 2 3 5 25 –2 6. t 1 4 5 25 –9 7. k 1 5

New Vocabulary • equivalent inequalities

56

23

What You’ll Learn• To use addition to solve

inequalities

• To use subtraction to solveinequalities

. . . And WhyTo solve a problem involvingsafe loads, as in Example 4

Part 1

are inequalities with the same solutions. For example,x 1 4 , 7 and x , 3 are equivalent inequalities.

You can add the same value to each side of an inequality, just as you did with equations.

Using the Addition Property of Inequality

Solve x 2 3 , 5. Graph the solution.

x 2 3 1 3 , 5 1 3 Add 3 to each side.

x , 8 Simplify.

Solve m 2 6 . 24. Graph your solution. m S 2; 0 1 2 3 4 5

11Quick Check�1 0 1 2 3 4 5 6 7 8 9 10

EXAMPLEEXAMPLE11

x + 4 < 7 x < 3

Equivalent inequalities

4-24-2

11 Using Addition to Solve Inequalities

Key Concepts Property Addition Property of Inequality

For every real number a, b, and c,if a . b, then a 1 c . b 1 c; if a , b, then a 1 c , b 1 c.

Examples 3 . 1, so 3 1 2 . 1 1 2. 25 , 4, so 25 1 2 , 4 1 2.

This property is also true for $ and #.

S SR

16



4-24-2

206

1. PlanObjectives1 To use addition to solve

inequalities2 To use subtraction to solve

inequalities

Examples1 Using the Addition Property

of Inequality2 Solving and Checking

Solutions3 Using the Subtraction

Property of Inequality4 Real-World Problem Solving

Math Background

Students may have used a balance scale to model equations.Here it may help them to solveinequalities if they refer to anunbalanced scale.






Adding Rational Numbers Lesson 2-1: Example 2Extra Skills and Word


Solving One-Step EquationsReview p. 118: Example 1

PowerPoint

Special NeedsTo help students understand infinity, use the numberline in Example 2. Ask: What number is to the right of20? 21 of 21? 22 Then ask: Will there ever be anumber that does not have yet another number to itsright? No

Below LevelPresent students with several pairs of inequalities andtheir solutions. Have students take turns explainingwhat was done to each inequality to produce itssolution.

L2L1

learning style: visual learning style: verbal

An inequality has an infinite number of solutions, so it is not possible to check allthe solutions. You can check your computations and the direction of the inequalitysymbol. The steps below show how to check that x < 8 describes the solutions toExample 1.

Step 1 Check the computation. See if 8 is the solution to the equation x 2 3 5 5.

x 2 3 5 5

8 2 3 0 5 Substitute 8 for x.

5 5 5 ✓

Step 2 Check the inequality symbol. Choose any number less than 8 andsubstitute it into x 2 3 , 5. In this case, use 7.

x 2 3 < 5

7 2 3 < 5 Substitute 7 for x.

4 < 5 ✓

Since the computation and the direction of the inequality symbol are correct,x 2 3 , 5 and x , 8 are equivalent inequalities. So the solution of x 2 3 , 5 is x , 8.

Solving and Checking Solutions

Solve 12 # x 2 5. Graph and check your solution.

12 1 5 # x 2 5 1 5 Add 5 to each side.

17 # x Simplify.

Check 12 5 x 2 5 Check the computation.

12 0 17 2 5 Substitute 17 for x.

12 5 12 ✓

12 # x 2 5 Check the direction of the inequality.

12 # 18 2 5 Substitute 18 for x.

12 # 13 ✓

Solve n 2 7 # 22. Graph and check your solution. n K 5;

Part 2

You can subtract the same number from each side of an inequality to create anequivalent inequality.

3 4 5 6 7 822Quick Check

10 200

EXAMPLEEXAMPLE22

12 Using Subtraction to Solve Inequalities

Key Concepts Property Subtraction Property of Inequality

For every real number a, b, and c,if a . b, then a 2 c. b2 c; if a , b, then a 2 c, b2 c.

Examples 3 . 21, so 3 2 2 . 21 2 2 25 , 4, so 25 2 2 , 4 2 2


Lesson 4-2 Solving Inequalities Using Addition and Subtraction 207

The word infiniteindicates that the numberof solutions is unlimited.The solutions cannot belisted.

Vocabulary Tip

For: Linear Inequalities ActivityUse: Interactive Textbook, 4-2

207

2. Teach

Guided Instruction

Tactile Learners

Let students use tiles to modelthe inequality. Have studentswrite the inequality sign on anindex card. Instruct them to placethe card between the two sets oftiles. Then have them solve theinequality. When the solution iscomplete, students can check bysubstituting tiles for a value of xin the original model.

Special Needs

Remind students that the dot andarrow in Example 2 means that 17and all numbers to the right of 17on the number line are solutionsto the inequality.

L1EXAMPLEEXAMPLE22

EXAMPLEEXAMPLE11

Advanced LearnersHave students discuss whether x � 1 is equivalent to-x � -1 and explain each step of their reasoning.

English Language Learners ELLBe sure students understand how the solution of anequation differs from the solution of an addition orsubtraction inequality. Ask them to compare and togive a real-world example of each.

L4



42 ± 65 ± b � 160, or b � 53

�1 0 1 2 63 4 5

17–19. See back of book.

Test-Taking Tip

1 A B C D E

2 A B C D E

3 A B C D E

4 A B C D E

5 A B C D E

B C D E

You can test inequalitychoices by substitutinga reasonable value forthe variable. If thevalue is NOT a solutionof the inequality youcan eliminate thatchoice as the correctanswer.

Using the Subtraction Property of Inequality

Solve y 1 5 , 27. Graph the solution.

y 1 5 2 5 , 27 2 5 Subtract 5 from each side.

y , 212 Simplify.

Solve t 1 3 $ 8. Graph and check your solution. t L 5;

You can use inequalities to model real-world situations.

Multiple Choice The maximum safe load of a chairlift is 680 lb. In the spring, acyclist and bicycle go to the top of the slope using the chairlift. The weight of theperson is 124 lb, and the weight of the bicycle is 32 lb. Which inequality bestdescribes how much additional weight w the chairlift could safely carry?

Relate plus

Define Let 5 the amount of weight that can be added to the chairlift.

Write 1

The inequality models the situation. So B is the correctanswer.

Your baseball team has a goal to collect at least 160 blankets for a shelter. Teammembers brought 42 blankets on Monday and 65 blankets on Wednesday. Write aninequality to describe how many blankets the team must donate on Friday to makeor exceed their goal.

State what number you would add to each side of the inequality to solve the inequality.

1. d 2 5 $ 24 5 2. 0 , c 28 8 3. z 2 4.3 $ 1.6 4.3

Solve each inequality. Graph and check your solution. 4–16. See margin.

4. x – 1 . 10 5. t – 3 , –2 6. –5 . b – 1 7. 7 # d 2 3

8. s 2 2 $ 26 9. r 2 9 # 0 10. 8 , n 2 2 11. 24 $ w 2 2

12. 21 , 24 1 d 13. y 2 # 25 14. . q 2 4 15. x 2 2 $ 0.5

16. 3.2 . 21.3 1 r 17. 23.4 . m 2 1.8 18. b 2 , 19. n 2 . 122

12

18

38

2 23

12

Examples 1, 2(pages 206, 207)

44Quick Check

124 1 32 1 w # 680

680#w124 1 32

w

safeload

is at mostadditional

weightweight of a person

and a bicycle

124 1 32 1 w $ 68032 1 w $ 680 1 124124 1 32 1 w # 680124 1 w # 680 1 32


33Quick Check0 6�6�12

EXAMPLEEXAMPLE33



GO forHelp

Additional Examples

Solve p - 4 � 1. Graph thesolution. p R 5

Solve 8 $ d - 2. Graph andcheck your solution. 10 L d,or d K 10

Guided Instruction

Diversity

Some students may have lived in ashelter or may live in one now. Besure other students are sensitivewhen discussing the Quick Checkquestion.

Additional Examples

Solve c + 4 � 7. Graph thesolution. c S 3

In order to receive a B in yourliterature class, you must earnmore than 350 points of readingcredits. Last week you earned 120 points. This week you earned90 points. How many more pointsmust you earn to receive a B? 141 points


Adapted Instruction

Closure

Ask students to compare solvinginequalities using addition andsubtraction with solvingequations using addition andsubtraction. In both cases, youmust add or subtract the sameamount to or from both sides.

L1

L3

44

�1 0 1 2 3 4 5

33

EXAMPLEEXAMPLE44

0 10 20

22

0 1 2 3 4 5 6

11

pages 208–211 Exercises

4. x S 11

5. t R 1�2�1 0 1 2

9 10 11 12 13 14

6. b R –4

7. d L 10

8 9 10 11 12 13

�7 �6 �5 �4 �3 �2

8. s L –4

9. r K 9

10. n S 10�5 0 5 10 15

9876 10 11

�6�7 �5�4 �3 �2

PowerPoint

PowerPoint

208


State what number you would subtract from each side of the inequality to solve the inequality.

20. w 1 2 . 21 2 21. 8 , 1 r 22. 5.7 $ k 1 3.1 3.1

Solve each inequality. Graph and check your solution. 23–38. See back of book.

23. w 1 4 # 9 24. m 1 5 . –3 25. 1 , 8 1 b 26. –2 $ 4 1 a

27. r 1 1 $ 25 28. k 1 3 # 4 29. 3 . 4 1 x 30. 25 , 1 1 p

31. 1 z $ 32. 7.5 1 y , 13 33. , m 1 2 34. 2.7 $ a 1 3

35. 22.9 , 4.1 1 p 36. $ h 1 37. 5.3 1 d . 3.8 38. t 1 ,

39. Vacation Budget Your brother has $2000 saved for a vacation. His airplaneticket is $637. Write and solve an inequality to find how much he can spend foreverything else. s ± 637 K 2000, $1363

40. Weekly Budget You have an allowance of $15.00 per week. You are in abowling league that costs $6.50 each week, and you save at least $5.00 eachweek. Write and solve an inequality to show how much you have left to spendeach week. s ± 6.50 ± 5 K 15, $3.50

41. Fund-Raising A school club is selling reflectors for Bicycle Safety Day. Eachmember is encouraged to sell at least 50 reflectors. You sell 17 on Monday and12 on Tuesday. How many reflectors do you need to sell on Wednesday to meetyour goal? r ± 17 ± 12 L 50, 21 reflectors

State what you must do to the first inequality in order to get the second.

42. 36 # 24 1 y; 40 # y 43. 9 1 b . 24; b . 15 44. m 2 , ; m ,

Solve each inequality.

45. w 2 3 1 1 $ 9 w L 11 46. 1 c # c K 3 47. y 2 0.3 , 2.8 y R 3.1

48. 26 . n 2 n R –5 49. z 1 4.1 , 25.6 50. 24.1 . y 2 0.9

51. 1 t 2 . 0 t S 52. 5 # v 2 4 2 7 v L 16 53. 3.6 1 k $ 24.5

54. 6 1 b 2 7 , 5 b R 6 55. m 1 2.3 # 21.2 56. 4 $ k 2 k K 4

57. h 2 $ 21 h L – 58. 27.7 $ x 2 2 x K –5.7 59. 22 . 9 1 3 1 w

60. Banking Your local bank offers free checking for accounts with a balance of atleast $500. Suppose you have a balance of $516.46 and you write a check for$31.96. How much must you deposit to avoid being charged a service fee?

61. a. If 45 1 47 5 t, does t 5 45 1 47? yesb. If 45 1 47 , r, is r , 45 1 47? noc. Discuss the differences between these two examples. See above left.

62. Gymnastics Suppose your sister wants to qualify for a regional gymnasticscompetition. At today’s competition she must score at least 34.0 points. Shescored 8.8 on the vault, 7.9 on the balance beam, and 8.2 on the uneven parallelbars. The event that remains is the floor exercise. a–b. See back of book.a. Write and solve an inequality that models the information.b. Explain what the solution means in terms of the original situation.c. Open-Ended Write three scores your sister could make that would allow her

to qualify for the regional gymnastics competition. Answers may vary.

12

12

34

34

16

56

23

45

15

3 12

12

78

38

12

Apply Your SkillsBB

Example 4(page 208)

2 18

38

34

14

122

25

35

53

53

Example 3(page 208)

Add 4 to each side. Subtract 9 from each side.

at least $15.50

61c. Answers may vary.Sample: The ≠ signmeans the sides areequal, so the twosides may beinterchanged. The Rsign does notindicate equality. Oneside cannot be bothgreater than and lessthan the other side.

y R –3.2

k L –8.1

m K –3.5

w R –14

Sample: 9.1, 9.2, 9.3

Add to each side.12


More than 71,000 athletescompete in gymnasticprograms in the United States.

z R –9.7

GPS


209

11. w K –2

12. d S 3

�1 0 1 2 3 4

�4�3�2�1 0 1

13. y K –4

14. q R 30 1 2 3 4 5

13

�6�5�4�3 1�2�1 0

12

15. x L 2.5

16. r R 4.5

�1 0 1 2 3 4 5

�1 0 1 2 3 4


A B 1-19, 42-54, 67-72

A B 20-41, 55-66, 73C Challenge 74-78



Error Prevention!

Exercises 4–19 Some studentsmay have difficulty deciding inwhich direction to draw the graphof a solution. Point out that whenthe solution has the variable onthe left, the graph points in thesame direction as the inequalitysymbol. When the solution has thevariable on the right, the graphpoints in the opposite direction ofthe inequality symbol.

Exercises 31–38 Suggest tostudents that they eliminate thefractions or decimals beforesolving.

2

1


Enrichment

Reteaching

Adapted Practice

Name Class Date

© P

ears

on E

duc

atio

n, In

c. A

ll rig

hts

rese

rved

.

Practice 4-2 Proportions and Similar Figures

Each pair of figures is similar. Find the length of x.

1. 2.

3. 4.

5. 6.

7. 8.

Use a proportion to solve.

9. nABC is similar to nXYZ. The length AB is 10. The length BC is 7.Find the length XY if the length YZ is 14.

10. Marty has a scale model of a car. The scale is 1 in. : 32 in. If the model is6.75 in. long, how long is the actual car?

11. A blueprint scale is 1 in. : 12 ft. The width of a building is 48 ft. What isthe width of the building on the blueprint?

12. Angie is using similar triangles to find the height of a tree. A stick thatis 5 ft tall casts a shadow that is 4 ft long. The tree casts a shadow that is22 ft long. How tall is the tree?

13. nABC is similar to nXYZ. The length AC is 10. The length BC is 16.What is the length XZ if the length YZ is 12?

14. A map has a scale of 1 in. : 25 mi. Two cities are 175 mi apart. How farapart are they on the map?

4 4

12x

4

3

4x

10

2

4

x

4

6

8

x

4.14.2

3.9

x

5

2.5

3x

107

6

x

15

12

9

x

Practice

L3

L4

L2

L1

L3


63. Computers Suppose your computer has nearly 64 megabytes (MB) of memory.Its basic systems require 12.8 MB. How much memory is available for otherprograms and functions? nearly 51.2 MB

64. To earn an A in Ms. Orlando’s math class, students must score a total of at least135 points on the three tests. On the first two tests, Amy’s scores were 47 and48. What is the minimum score she must get on the third test to earn an A?

65. a. Open-Ended Use each of the inequality symbols ,, #,., and $ to writefour addition or subtraction inequalities. a–b. Check students’ work.

b. Solve each of the inequalities in part (a) and graph your solution.

66. a. Sam says that he can solve z2 8.6$ 5.2 by replacing z with 13, 14, and 15.When z5 13, the inequality is false. When z5 14 and z5 15, the inequality istrue. So Sam says that the solution is z$ 14. Is his reasoning correct? Justifyyour answer. No; the solution is z L 13.8, so z L 14 is not correct.

b. Critical Thinking Explain why substituting values into the inequality doesnot guarantee that your solution is correct. See margin.


67. 4x 1 4 2 3x $ 5 x L 1 68. 25n 2 3 1 6n , 2 n R 5

69. 7t 2 (6t 2 2) # 21 t K –3 70. 5k 2 2(2k 1 1) . 8 k S 10

71. 26(a 1 2) 1 7a # 12 a K 24 72. 22(a 2 3) 1 3(a 1 2) , 4 a R –8

73. Geometry The Triangle Inequality Theorem states that the sum of the lengthsof any two sides of a triangle is greater than the length of the third side.Following are inequalities for sides of the triangle shown.

a 1 b . c b 1 c . a a 1 c . b

a. Write an inequality using c 2 b and a. a S c – bb. Write an inequality using a 2 c and b. b S a – cc. Write an inequality using b 2 a and c. c S b – ad. Writing Write a generalization about the

length of the third side and the difference of the lengths of the other two sides.

Reasoning Decide if each inequality is true for all real numbers. If the inequality isnot true, give a counterexample.

74. a 2 b , a 1 b See above left. 75. If a $ b, then a 1 c $ b 1 c. true

76. If c . d, then a 2 c , a 2 d. true 77. If a , b, then a , b 1 c.

78. Reasoning Find real numbers x, y, z, and w for which it is true that x . y and z . w, but it is not true that x 2 z . y 2 w.

79. Solve x 1 5 , 13. BA. x . 8 B. x , 8 C. x . 18 D. x , 18

80. Solve -12 1 n . 20. GF. n , 32 G. n . 32 H. n , 8 J. n . 8

Multiple Choice

ChallengeCC

a

c

b

Test Prep


In 1971, a computer chipcould hold 2300 transistors. In 2004, a chip could hold410,000,000 transistors.

See above left.

See left.

77. not true; samplecounterexample: for a ≠ 0, b ≠ 1, and c ≠–2, 0 R 1 and 0 � 1 ± (–2)

73d. The length of the thirdside must be greaterthan the difference ofthe lengths of the othertwo sides.

74. not true; samplecounterexample: for a ≠ 5 and b ≠–6, 5 –(–6) � 5 ±(–6)

Answers may vary. Sample:For x ≠ 2, y ≠ 1, z ≠ 4, and w ≠ 3, 2 S 1 and 4 S 3, but 2 – 4 � 1 – 3.

GO nlineHomework HelpVisit: PHSchool.comWeb Code: ate-0402

40 points

210

Lesson Quiz

Solve each inequality. Graph thesolutions.

1. p - 7 $ -5 p L 2

2. w - 3 � -9 w R –6

3. x + 6 � 4 x S –2

4. 13 $ 9 + h 4 L h, or h K 4


Group students in pairs. Instructeach student to model aninequality using tiles and an indexcard with the inequality signwritten on it. Have studentsexchange inequalities with theirpartners and find the solutions.Encourage students to check theirsolutions.

66b. Answers may vary.Sample: Substitutingvalues does not workbecause there is alwaysthe possibility that thesolution lies betweenvalues that make theinequality true and avalue that does not.

�1 0 1 2 3 4 5

�3�2�1 0 1 2 3

�4�3�2�1�5�6�7

�1�2 0 1 2 3 4

PowerPoint

4. Assess & Reteach

81. Which graph represents all real number solutions of x 1 4 $ 8? AA. B.

C. D.

82. Which of the following is a solution for 5 , n 2 0.1? JF. 4.99 G. 5.01 H. 5.10 J. 5.11

83. Hector is flying his plane. To avoid a storm, he climbs 5500 ft withoutgoing above his plane’s maximum safe altitude of 35,000 ft. Theinequality a 1 5500 # 35,000 represents his original altitude a in feet.Which of the following could have been the original altitude? DA. 40,500 ft B. 30,000 ft C. 29,750 ft D. 27,750 ft

84. The leading scorer in your high school’s basketball division finished theseason with a game average of 20 points for 25 games. As the division’ssecond leading scorer, you have a 19.5 point per game average for 24games. You have your last game yet to play.

How many points must you score in the final game of the season toovertake the division’s leading scorer? Show your work. See above left.

Mixed Review

Define a variable and write an inequality to model each situation.

85. An octopus can be up to 10 ft long. Let c ≠ length of octopus in feet. c K 10

86. A hummingbird migrates more than 1850 mi. Let h ≠ distance in miles

87. Your average in algebra class must be 90 or greater for you to receive an A for the term. Let a ≠ average. a L 90

88. You must read at least 25 pages this weekend.

Simplify.

89. 92 1 17 98 90. 4(5 2 3)2 2 32 7 91. 0.2(4.2 2 3.4) 1 0.4

92. 3 ? 2 1 52 31 93. 3 1 72 2 4 48 94. 43 1 32 73

95. 6(5 2 2)2 1 4 58 96. 2 1 8(6 1 22) 82 97. 33 2 23 1 7 26

Lesson 1-2

Lesson 4-1

Short Response

�3 �2 �1 0 1 2 3�1 0 1 2 3 4 5

�5 0 1�4 �3 �2 �1�1 0 1 2 3 4 5


a hummingbird migrates. h S 1850

Let p ≠ number of pages to read. p L 25

0.56

84. [2] at least 33 points;24(19.5) ± x S 25(20)

468 ± x S 500x S 32

(OR equivalentexplanation)

[1] incorrect answer ORno work or explanation

Algebra at WorkAlgebra at Work

Marketing directors rely on equations and inequalities to predict theactions their companies must take to stay competitive. For example, themarketing director of a manufacturing company determines how much the cost of raw materials can increase before the company must raise theprice of its finished goods or services. The director also predicts the effect ofprice changes on the quantity of goods and services sold by his company.

Marketing Director


PHSchool.com

For: Information about a career in marketingWeb Code: atb-2031

GO forHelp

211

Test Prep


Transparencies

Exercise 81 Point out to studentsthat they can quickly eliminateanswer choices B and D, sincethese have open dots.


Review page 118 and Lesson 3-1


1. 8 5 t 16 2. 14 5 221x – 3. 5 21 –6

4. 5d 5 32 6.4 5. x 5 212 –18 6. 0.5n 5 9 18

Write an inequality for each graph.

7. x K –1 8. x S 3�1 0 1 2 3 4 5�3 �2 �1 0 1 2 3

23

x6

23

12

What You’ll Learn• To use multiplication to

solve inequalities

• To use division to solveinequalities

. . . And WhyTo find how much food canbe purchased for a food bank,as in Example 4

Part 1 Using Multiplication to Solve Inequalities

You can multiply each side of an inequality by the same number, just as you didwith equations. When you multiply each side of an inequality by a positive number,the direction of the inequality symbol stays the same. When you multiply each sideby a negative number, the direction of the inequality symbol reverses.

4-34-3 Solving Inequalities Using Multiplication and Division

11 Using Multiplication to Solve Inequalities

Activity: Multiplying Each Side of an Inequality

Consider the inequality 4 . 1.

1. Copy and complete each statement at the rightby replacing each ■ with ,, ., or 5.

2. What happens to the inequality symbol whenyou multiply each side by a positive number?

3. What happens to the inequality symbol whenyou multiply each side by zero?

4. What happens to the inequality symbol whenyou multiply each side by a negative number?

4 ? 3 1 ? 3■

4 ? 2 1 ? 2■

4 ? 1 1 ? 1■

4 ? 0 1 ? 0■

4 ? �1 1 ? �1■

4 ? �2 1 ? �2■

4 ? �3 1 ? �3■

Key Concepts Property Multiplication Property of Inequality for c > 0

For every real number a and b, and for c. 0,

if a . b, then ac . bc; if a , b, then ac , bc.

Examples 4 . 21, so 4(5) . 21(5). 26 , 3, so 26(5) , 3(5).


There is no change in the inequality sign.

The inequality becomes an equality: 0 ≠ 0.

The inequality sign is reversed.

S

S

S

≠

R

R

R


4-34-3

212

1. PlanObjectives1 To use multiplication to solve

inequalities2 To use division to solve

inequalities

Examples1 Multiplying by a

Positive Number2 Multiplying by a

Negative Number3 Dividing to Solve

an Inequality4 Real-World Problem Solving

Math Background

Given their great usefulness indeveloping insight, patternsshould be discussed as often aspossible.






Solving One-Step Equations Review p. 118: Example 2

Solving Two-Step EquationsLesson 3-1: Example 4Extra Skills and Word


PowerPoint

Special NeedsSome students may be confused about when toreverse the inequality symbol. Have them alwayscheck their answers to make sure the direction of theinequality symbol is correct.

Below LevelHave students pick integer solutions to Example 3from the graph and substitute them in the originalinequality to verify solutions.

L2L1

learning style: verbal learning style: verbal

Lesson 4-3 Solving Inequalities Using Multiplication and Division 213

You can use the Multiplication Property of Inequality to solve inequalities thatinvolve division.

Multiplying by a Positive Number

Solve , 21. Graph and check the solution.

2 , 2(21) Multiply each side by 2. Do not reverse the inequality symbol.

x , 22 Simplify each side.

Check 5 21 Check the computation.

0 21 Substitute 22 for x.

21 5 21 ✓ Simplify.

, 21 Check the direction of the inequality.

, 21 ✓ Substitute 23 for x.

Solve each inequality. Graph and check your solution.

a. . b S 2; b. $ d L 2 ; c. # 23 y K –1.5;

Multiplying by a Negative Number

Multiple Choice Which graph shows the solution of ?

$ 2 Multiply each side by , the reciprocal of . Reverse the inequality symbol.

n $ 23 Simplify.

B is the correct answer.

Solve each inequality. Graph and check the solution. See back of book for graphs.

a. . 2 1 k R 4 b. 2t , t S – c. 6 # w K –102 35w1

2122

k4

22Quick Check

�3 �2 �1 0 1 2 3�4 4�5�6

2 232

32Q232RQ22

3 nRQ232R

6�4 �3 �1�2 0 21 3 4 5

�6 �4�5 �3 �1�2 0 21 3 4

�6 �4�5 �3 �1�2 0 21 3 4

�6 �4�5 �3 �1�2 0 21 3 4

223n # 2

EXAMPLEEXAMPLE22

y0.5

12

56

d3

12

b4

11Quick Check

2 32

x2

2 22

x2

�1�2�3�4�5 0 1 2 3 4 5

Qx2R

x2

EXAMPLEEXAMPLE11

Key Concepts Property Multiplication Property of Inequality for c < 0

For every real number a and b, and for c, 0,

if a . b, then ac , bc; if a , b, then ac . bc.

Examples 4 . 21, so 4(22) , 21(22). 26 , 3, so 26(22) . 3(22).


�1 0 1 2 3 �1 0 1 2 3 4 �3�2�1 0 1

Test-Taking Tip

1 A B C D E

2 A B C D E

3 A B C D E

4 A B C D E

5 A B C D E

B C D E

You can use the endpointand another point on agraph to determine if thegraph shows the solutionof an inequality. Use theendpoint to check forequality. For the answerB, –3 is the endpoint and0 is another point.

✓

✓

This confirms that B isthe graph of thesolution of .22

3(n) # 2

223(0) # 2

223(23) 5 2

213

2. Teach

Guided Instruction

ActivityHave students repeat theInvestigation for an inequalitythat contains negative numbersonly, such as -3 � -1. Then havethem contrast and compare theanswers for both inequalities. Thiswill help students realize that thereversal of the inequality signdoes not depend on positive ornegative numbers in theinequality.

Error Prevention

Students may think that since a negative is part of themultiplication, they shouldreverse the inequality sign. Stress that you reverse theinequality sign only if the number by which you aremultiplying or dividing both sidesis negative.

Alternative Method

Show students that 0 is theeasiest number to use to check a solution. Since 0 is part of the solution in Example 2,demonstrate how students canmentally replace n with 0, thusmaking the right side of theinequality 0. Two is greater than0, so the solution is correct. Zerocan be used whether it is part ofthe solution set or not. If it is notpart of the solution set, then thechecking process will end in afalse statement.

In Example 3, substituting 0 for zleads to the false inquality 0 # -5;so 0 is not in the solution set.

EXAMPLEEXAMPLE22

EXAMPLEEXAMPLE11

Advanced LearnersDraw a chart on the board similar to the one on p. 212. Begin with 4 � 3. Help students decide whichsymbol replaces each box.

English Language Learners ELLVerify that students understand that reversing theinequality symbol means changing its direction. Havethem check their answers to confirm theirunderstanding.

L4



Part 2

Solving inequalities using division is similar to solving inequalities usingmultiplication. Remember that division by zero is undefined.

Dividing to Solve an Inequality

Solve 25z $ 25. Graph the solution.

z # Divide each side by 25. Reverse the inequality symbol.

z # 25 Simplify.

Solve the inequality. Graph and check your solution.a. 22t , 28 t S 4; b. 23w $ 12 w K –4; c. 0.6 . 20.2n n S –3;

There are times when you must think about which types of numbers are acceptableas solutions of inequalities that represent real-world situations.

Community Service The student council votes to buy food for a local food bank.A case of 12 jars of spaghetti sauce costs $13.75. What is the greatest number ofcases of sauce the student council can buy if they use at most $216 for this project?

Relate times

Define Let 5 the number of cases of spaghetti sauce.

Write ?

13.75c# 216

# Divide each side by 13.75.

c# 15.71 Simplify and round to the nearest hundredth.

The student council does not have enough money to buy 16 cases, so they can buyat most 15 cases of sauce for the food bank.

21613.75

13.75c13.75

216#c13.75

c

total costis at mostthe numberof cases

cost per case


33Quick Check0 1�9 �8 �7 �6 �5 �4 �3 �2 �1

2525

2525

EXAMPLEEXAMPLE33

12 Using Division to Solve Inequalities

Key Concepts Property Division Property of Inequality

For every real number a and b, and for c . 0,

if a . b, then . ; if a , b, then , .

Examples 6 . 4, so . . 2 , 8, so , .

For every real number a and b, and for c , 0,

if a . b, then , ; if a , b, then . .

Examples 6 . 4, so , . 2 , 8, so . .

This property also applies to $ and #.

82 2

22 2

42 2

62 2

bc

ac

bc

ac

82

22

42

62

bc

ac

bc

ac


Careers The duties of amanager of a nonprofitorganization, such as a foodbank, include organizing andsupervising volunteers.

�2 0 2 4 6 8 �2�3�4�5�6�7 �4�3�2�1 0 1

For: Solving Inequalities Activity Use: Interactive Textbook, 4-3

Additional Examples

Solve � -2. Graph and checkthe solution. z S –6

Solve 3 # - x. Graph andcheck the solution. –5 L x, or x K –5

Guided Instruction

Teaching Tip

Instruct students to graph c # 15.71. Tell them to place the dot on the left side close to,but not on, 16. Ask students ifevery number graphed canrepresent a case of food. Havethem explain their answers. Only whole numbers can expressthe number of cases, becauseyou can only buy a whole case,not part of one.

Additional Examples

Solve -4c � 24. Graph thesolution. c S –6

Your family budgets $160 tospend on fuel for a trip. Howmany times can they fill the car’sgas tank if it costs $25 each time?6 times


Adapted Instruction L1

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Closure

Have students state the maindifference between solvinginequalities using multiplicationand division and solvingequations using multiplication

and division. Both cases requirethe same operations. However, insolving inequalities in which youhave to multiply or divide eachside by a negative number, youmust reverse the direction of theinequality.


1. t L –4;

2. s R 6;�2 0 2 4 6 8

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PowerPoint

PowerPoint

214

Students in the school band are selling calendars. They earn $.40 on each calendarthey sell. Their goal is to earn more than $327. Write and solve an inequality to findthe fewest number of calendars they can sell and still reach their goal.

EXERCISES For more practice, see Extra Practice.


Solve each inequality. Graph and check your solution.

1. $ 21 2. , 1 3. 1 # 4. 2 ,

5. 22 , 6. $ 0.5 7. 4 . x 8. 25 # k

9. 0 , 2 x 10. y $ 0 11. 2 x . 25 12. 6 $ d

13. , c 14. b $ 15. 2 u . 16. 2 . n

17. 3t , 29 18. 4m $ 8 19. 10 # 22w 20. 220 . 25c

21. 227 $ 3z 22. 27b . 42 23. 18d , 212 24. 23x # 16

25. 27 , 2q 26. 16 . 3.2h 27. 21.5d , 26 28. 3.6 # 20.8m

29. Fund-Raising The science club charges $4.50 per car at their car wash. Writeand solve an inequality to find how many cars they have to wash to earn at least $300. 4.5c L 300, 67 cars

30. Earnings Suppose you earn $6.15 per hour working part time at a dry cleaner.Write and solve an inequality to find how many full hours you must work toearn at least $100. 6.15h L 100, 17 hours

Write four solutions to each inequality. 31–38. Answers may vary. Samples given.

31. # -1 32. $ -4 33. -1 $ 34. 0.5 . c

35. q . 4 36. 1 , 2 s 37. 24.5 # 20.9p 38. 22.7w $ 28

Tell what you must do to the first inequality in order to get the second.

39. . 3; c , 212 40. # 22; n# 210

41. 5z .225; z . 25 42. b # 3; b # 4

43. 212 , 4a; 23 , a 44. 2b $ 3.4; b # 23.4

Replace each ■ with the number that makes the inequalities equivalent.

45. ■ s . 14; s , 27 –2 46. ■ x $ 25; x # 25 –5

47. 28u # ■ ; u $ 20.5 4 48. 22a . ■ ; a , 29 18

49. 36 , ■ r; r , 23.6 –10 50. 2k # ■ ; k $ 27.5 7.5

51. Critical Thinking If x $ y and 2x $ 2y, what can you conclude about x and y?

Estimation Estimate the solution of each inequality. 52–55. Estimates may vary.

52. 2 2.099r , 4 53. 3.87j . 224 54. 20.95 $ p 55. s # 2142 2039

12

34

n52

c4

572

34

12

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Apply Your SkillsBB

Example 4(page 214)

Example 3(page 214)

2 56

58

56

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34

232

49

2 32

57

43

78

52

232

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(page 213)

44Quick Check




0.4c S 327; 818 calendars

–2, –3, –4, –5 –12, –10, –8, 0 –3, –4, –5, –6 , 0, –1, –212

–10.4, –11, –12, –13

39–44. See left.

x and y are equal.

r S –2 j S –6 p K 42 s L 28

39. Multiply each side by–4 and reverse theinequality symbol.

40. Multiply each side by 5.

41. Divide each side by 5.

42. Multiply each side by .

43. Divide each side by 4.

44. Multiply each side by–1 and reverse theinequality symbol.

43

5, 4, 3, 2–2, –3, –4, –5–6, –7, –8, –9


GO forHelp

1–28. See marginpp. 215–217.

215

3. w K –2;

4. p R –8;

5. y S –4;

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A B 1-16, 31-36, 39-42, 54-55, 67-75

A B 17-30, 37-38, 43-53,56-66, 76-77

C Challenge 78-83



Exercises 31–38 Encouragestudents to use some numbersthat are not integers.

Exercise 77 Suggest to studentsthat they begin by comparing the correct solution with Kia’ssolution.

2

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Enrichment

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Adapted Practice

Name Class Date

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Practice 4-3 Proportions and Percent Equations

Solve each problem.

1. 25% of what is 28? 2. What percent of 72 is 18? 3. 60% of what is 45?

4. What percent of 12 is 6? 5. What is 60% of 12? 6. 75% of what is 48?

7. What is 20% of 650? 8. What percent of 150 is 90? 9. What percent of 90 is 63?

10. What is 38% of 60? 11. 22.5% of what is 42? 12. 45% of what is 99?

13. What percent of 210 is 10.5? 14. 160% of what is 124? 15. What is 39% of 1500?

16. What is 250% of 14? 17. What percent of 20 is 36? 18. What is 8.25% of 160?

Write an equation to model each question and solve.

19. Pablo has a goal to lose 25 lb. He has lost 16 lb. What percent of his goalhas he reached?

20. You spent 16% of your vacation money on food. If you spent $48 onfood, how much money did you spend on your vacation?

21. A writer earns $3400 a month. Last month she spent $204 on food.What percent of her income was spent on food?

22. Kiko spends 30% of her monthly income on rent. If she pays $810 forrent each month, what is her monthly income?

23. Suppose that 62.5% of freshmen entering a college graduate from it. Ifthere are 2680 freshmen, how many will graduate from that college?

The formula for determining simple interest is I ≠ prt. Using this formula,solve the following problems.

24. You invest $1500 for three years. Find the amount of simple interest youearn at an annual rate of 8.25%.

25. Suppose you invested $1200 for four years. You earned $312 in simpleinterest. What is the interest rate?

26. Suppose you invested some money at 8% simple interest for five years.If you received $500 in interest, how much money did you invest?

Write an equation to model each question and solve.

27. What is 7% of 480? 28. What percent of 80 is 48? 29. 90% of what is 27?

30. What is 150% of 26? 31. 125% of what is 175? 32. What is 10.25% of 280?

33. What is 35% of 360? 34. What percent of 36 is 9? 35. 75% of what is 90?

36. 45% of what is 36? 37. What is 80% of 120? 38. What percent of 20 is 8?

39. 25% of what is 92? 40. What percent of 30 is 90? 41. What is 39% of 800?

Practice

L3

L4

L2

L1

L3


56. Safe Load An elevator like the one at the left can safely lift at most 4400 lb. Aconcrete block has an average weight of 42 lb. What is the maximum number ofconcrete blocks that the elevator can lift? 104 concrete blocks

57. Writing Explain how solving the equation 5 4 is similar to and differentfrom solving the inequality . 4. See margin p. 151.

58. Open-Ended Write four different inequalities with x . 3 as their solution thatyou can solve using multiplication or division. Answers may vary.


59. 4d # 228 60. . 5 u S 35 61. 2 , 28s s R – 62. k $ 245

63. 0.3y , 2.7 y R 9 64. 9.4 # 24t 65. 2h $ 4 h K –4 66. x . 5 x S 2

67. 24 , 68. 0 , b R 0 69. . 2 p 70. 20.2m $ 9.4

71. 6 , 29g g R – 72. 4n $ 9 n L 2 73. 23.5 , 2m 74. z $ 21

75. Michael solved the inequality 22 . and got 6 , y. Erica solved the sameinequality and got y . 6. Are they both correct? Explain.

76. A friend calls you and asks you to meet at a location 3 miles from your home in20 minutes. You set off on your bicycle after the telephone call. Write and solvean inequality to find the average rates in miles per minute you could ride to beat your meeting place within 20 minutes. R r; r S 0.15

77. a. Error Analysis Kia solved 215q # 135 by adding 15 to each side of theinequality. What mistake did she make? See left above.

b. Kia’s solution was q # 150. She checked her work by substituting 150 for qin the original inequality. Why didn’t her check let her know that she hadmade a mistake? 150 satisfies the original inequality –15q K 135.

c. Open-Ended Find a number that satisfies Kia’s solution but does not satisfythe original inequality. Answers may vary. Sample: –15

Reasoning If a, b, and c are real numbers, for which values of a is each statement true?

78. If c , 0, then ac , a. a S 0 79. If b . c, then ab . ac. a S 0

80. If b . c, then a2b . a2c. a u 0 81. If b . c, then , . a R 0

82. Packaging Suppose you have a plastic globe that you wish to put into a giftbox. The circumference of the globe is 15 in. The edges of cube-shaped boxesare either 3 in., 4 in., 5 in., or 6 in. Write and solve an inequality to find theboxes that will hold the globe. (Hint: circumference5 p ? diameter)

83. Tiling a Floor The Sumaris’ den floor measures 18 ft by 15 ft. They want tocover the floor with square tiles that are ft2. Write and solve an inequality tofind the least number of tiles they need to cover the floor.


84. Solve = 16. 40

85. Mr. Houston expects to pay $16,800 in income taxes. This is no more than of his salary. What is his least possible earned income? 504001

3

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Test Prep


Depending on its size, anelevator at a construction sitecan have a maximum loadfrom 900 lb to 20,000 lb.

k L –30

x R –9 p S –212

m R 3.5 z L –212

m K –47

Yes; in each case, y is greater than 6.

d S 15, 5-in. box or 6-in. box π

x L 18(15), 480 tiles916

77a. She should havedivided each sideby –15.

t K –2.35

Sample: 2x S 6, x S 4, –x R –3, – R –35

x5

43

d K –7

GPS

216


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17. t R –3;

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22. b R –6;

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23. d R – ;

24. x L –5 ;

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Lesson Quiz

Solve each inequality. Graph thesolution.

1. $ -3 y L –6

2. - � -1 p S 3

3. 6x � 30 x R 5

4. 48 $ -12h –4 K h,or h L –4


Organize students into pairs. Haveeach pair complete Exercise 76.Then ask each pair to solve the inequalities written byanother pair.

Test Prep

A sheet of blank grids is availablein the Test-Taking Strategies withTransparencies booklet. Give thissheet to students for practice withfilling in the grids.


Transparencies

Exercise 89 Remind students tocheck for comparable units ofmeasure. Be sure that the mileand the length of a can areconverted to the same unit ofmeasure.

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4. Assess & Reteach

217


86. Five boxes of tiles will cover 30 ft2. What is the least number of boxes oftiles needed to tile 320 ft2? 54

87. What is the greatest number of 37¢ stamps you can buy for $5.00? 13

88. The length of a rectangle is 90 in. Its area is less than 380 in2. What is thegreatest possible width of the rectangle, to the nearest inch? 4

89. The Environmental Club of City Middle School set a goal of collecting “onemile of cans.” If the height of a typical 12 oz aluminum can is in., what isthe fewest number of cans needed to reach their goal? 13339

Mixed Review


90. x 1 3 # 24 91. . 1 w 92. t 2 3.4 $ 5.8 93. 0 , d 2 4

94. 3.2 $ m 1 7.1 95. y 2 2 , 6 96. 23 . a 2 4 97. k 1 12 # 15


98. 3t 1 5 2 t 5 9 2 99. 23(2n 1 1) 5 9 100. 2 5 1

Name the property that each exercise illustrates. 101–106. See left.

101. (27) 1 7 5 0 102. 23 5 1 103. 2 1 0 5 2

104. 1x 5 x 105. 4(2 ? 5) 5 (4 ? 2)5 106. 4(2 ? 5) 5 4(5 ? 2)

Checkpoint Quiz 1 Lessons 4-1 through 4-3

Solve each inequality. Graph your solution. 1–6. See left for graphs.

1. 6 , c 1 1 c S 5 2. 5x , 230 x R –6 3. # 22 p K –6

4. y 2 4 $ 22 y L 2 5. 12 1 g , 4 g R –8 6. 23b $ 15 b K –5

7. Determine whether each of the following is a solution of x 1 7 # 3.

a. 24 yes b. 0 no c. 2 yes d. 23.9 no

8. Determine whether each of the following is a solution of 24x , 212.

a. 3 no b. 0 no c. no d. p yes

Write and solve an inequality that models each situation.

9. You plan to buy a bicycle that will cost at least $180. You have saved $38 andyour parents have given you $50.a. Write an inequality to find how much more money m you need to save.b. Solve your inequality. a. m ± 38 ± 50 L 180 b. m L 92

10. Your local garden shop has plants on sale for $1.50 each. You are planning avegetable garden. You have $20 to spend on tomato plants.a. Write an inequality to find the greatest number of plants p you can buy.b. How many plants can you buy? a. 1.50p K 20 b. 13 plants

73

174

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14

12

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34

58

Lesson 4-2

434



x K –7 w R –18 t L 9.2

101. Prop. of Opposites102. Prop. of Reciprocals103. Ident. Prop. of Add.104. Ident. Prop. of Mult.105. Assoc. Prop. of Mult.106. Comm. Prop. of Mult.

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28. m K –4.5;

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and the inequality, youmultiply each side by –3.For the inequality, youmust reverse theinequality symbol.

Use this Checkpoint Quiz to checkstudents’ understanding of theskills and concepts of Lessons 4-1through 4-3.

ResourcesGrab & Go• Checkpoint Quiz 1

Lesson 4-4 Solving Multi-Step Inequalities 219

Solving Multi-Step Inequalities

Part 1 Variables on One Side

Sometimes you need to perform two or more steps to solve an inequality.Models can help you understand how to solve multi-step inequalities.

The tiles model the inequality.

Add 3 to each side.

Simplify by removing the zero pairs.

Divide each side into two equalgroups.

Each green tile is less than twoyellow tiles, so x R 2.

< x , 2

< 2x2 , 4

2

<2x , 4

<2x 2 3 1 3 , 1 1 3

<2x 2 3 , 1

4-44-4

What You’ll Learn• To solve multi-step

inequalities with variableson one side

• To solve multi-stepinequalities with variableson both sides

. . . And WhyTo find the measurements ofa banner, as in Example 2

Lessons 3-1 and 3-2

Solve each equation, if possible. If the equation is an identity or if it has nosolution, write identity or no solution.

1. 3(c 1 4) 5 6 –2 2. 3t 1 6 5 3(t 2 2) no solution

3. 5p 1 9 5 2p 2 1 –3 4. 7n 1 4 2 5n 5 2(n 1 2)

5. k 2 1 k 5 1 6. 2t 2 32 5 5t 1 1 –11

Find the missing dimension of each rectangle.

7. 8. perimeter = 78 in.

26 in.

w

perimeter = 110 cm

15 cm

�

29

76

23

12

13

11 Solving Inequalities With Variables on One Side

identity

40 cm 13 in.


4-44-4

219

1. PlanObjectives1 To solve multi-step

inequalities with variables onone side

2 To solve multi-stepinequalities with variables onboth sides

Examples1 Using More Than One Step2 Real-World Problem Solving3 Using the Distributive

Property4 Gathering Variables on One

Side of an Inequality5 Multi-Step Inequalities

Math Background

Many real world examples aremodeled by these more complexinequalities. Students should haveseveral different ways of solvingmany exercises.






Solving Two-Step Equations Lesson 3-1: Example 1Extra Skills and Word


Solving Multi-Step EquationsLesson 3-2: Examples 3, 4Extra Skills and Word


PowerPoint

Special NeedsHave students solve multi-step inequality problems inthe same way as they would multi-step equations.Point out that they will only change the inequalitysymbol if they multiply or divide by a negativenumber.

Below LevelAs the inequalities become more complex, reassurestudents that, just like multi-step equations, multi-stepinequalities can be solved by performing one step at a time.

L2L1



To solve inequalities, you undo addition and subtraction first. Then undomultiplication and division.

Using More Than One Step

Solve 7 1 6a . 19. Check the solution.

7 1 6a 2 7 . 19 2 7 Subtract 7 from each side.

6a . 12 Simplify.

. Divide each side by 6.

a . 2 Simplify.

Check 7 1 6a 5 19 Check the computation.

7 1 6(2) 0 19 Substitute 2 for a.

19 5 19 ✓

7 1 6a . 19 Check the direction of the inequality.

7 1 6(3) . 19 Substitute 3 for a.

25 . 19 ✓

Solve each inequality. Check your solution.a. 23x 2 4 # 14 x L –6 b. 5 , 7 2 2t t R 1 c. 28 , 5n 2 23 n S 3

You can adapt familiar formulas like the formula for the perimeter of a rectangleto write inequalities. You determine which inequality symbol to use from the real-world situation.

Geometry The school band needs a banner to carry in a parade. The bannercommittee decides that the length of the banner should be 18 feet. A committeemember drew the diagram at the left to help understand the problem. What are thepossible widths of the banner if they can use no more than 48 feet of trim?

Relate Since the border goes around the edges of a rectangular banner, you canadapt the perimeter formula P 5 2O 1 2w.

plus

Write 1

2(18) 1 2w # 48

36 1 2w # 48 Simplify 2(18).

36 1 2w 2 36 # 48 2 36 Subtract 36 from each side.

2w # 12 Simplify.

# Divide each side by 2.

w # 6 Simplify.

The banner’s width must be 6 feet or less.

To make a second banner, the committee decided to make the length 12 feet. Theyhave 40 feet of a second type of trim. Write and solve an inequality to find thepossible widths of the second banner.

22Quick Check

122

2w2

48#2w2(18)

the lengthof trim

can be nomore than

twice thewidth

twice thelength


11Quick Check

126

6a6

EXAMPLEEXAMPLE11

2(12) ± 2w K 40, so the banner’s width must be 8 feet or less.

220

2. Teach

Guided Instruction

Math Tip

Point out to students that theyare performing the order ofoperations in reverse to isolatethe variable on one side of theequation.

Teaching Tip

Ask students to list somereasonable widths for the bannerand some unreasonable widths,such as 6 inches, that meet thecriteria of the solution.

Visual Learners

Some students may forget todistribute the factor 2 (outsidethe parentheses) to the 2 insidethe parentheses, and somestudents may distribute it to the3t. Have students draw arrowsfrom the factor 2 to both termsinside the parentheses.

Additional Examples

Solve 5 + 4b � 21. Check thesolution. b R 4

The band is making arectangular banner that is 20 feetlong with trim around the edges.What are the possible widths thebanner can be if there is no morethan 48 feet of trim? 4 feet orless

Solve 3x + 4(6 - x) � 2. x S 2233

22

11

EXAMPLEEXAMPLE33

EXAMPLEEXAMPLE22

EXAMPLEEXAMPLE11

Advanced LearnersHave students write a multi-step inequality withvariables on both sides and that requires the DistributiveProperty. Let them exchange inequalities and solve.

English Language Learners ELLBe sure that students understand the word problems,as in Example 2. Have them explain to a partner whatthey need to do, and give the equivalent mathsentence.

L4


PowerPoint


Using the Distributive Property

Solve 2(t 1 2) 2 3t $ 21.

2t 1 4 2 3t $ 21 Use the Distributive Property.

2t 1 4 $ 21 Combine like terms.

2t 1 4 2 4 $ 21 2 4 Subtract 4 from each side.

2t $ 25 Simplify.

# Divide each side by 21. Reverse the inequality symbol.

t # 5 Simplify.

Solve each inequality. Check your solution.a. 4p 1 2(p 1 7) , 8 b. 15 # 5 2 2(4m 1 7) c. 8 . 3(5 2 b) 1 2

Part 2 Variables on Both Sides

Many inequalities have variables on both sides of the inequality symbol. You needto gather the variable terms on one side of the inequality and the constant terms onthe other side.

Gathering Variables on One Side of an Inequality

Solve 6z 2 15 , 4z 1 11.

6z 2 15 2 4z , 4z 1 11 2 4z To gather variables on the left, subtract 4z from each side.

2z 2 15 , 11 Combine like terms.

2z 2 15 1 15 , 11 1 15 To gather the constants on the right, add 15 to each side.

2z , 26 Simplify.

, Divide each side by 2.

z , 13 Simplify.

Solve 3b 1 12 . 27 2 2b. Check your solution. b S 3

Multi-Step Inequalities

Gridded Response Solve 23(4 2 m) $ 2(4m 2 14).

212 1 3m $ 8m 2 28 Use the Distributive Property.

212 1 3m 2 8m $ 8m 2 28 2 8m Subtract 8m from each side.

212 2 5m $ 228 Combine like terms.

212 2 5m 1 12 $ 228 1 12 Add 12 to each side.

25m $ 216 Simplify.

#Divide each side by 25. Reverse the inequality symbol.

m # Simplify.

Solve 26(x 2 4) $ 7(2x 2 3). Check your solution. x K 214

55Quick Check231

5

21625

25m25

EXAMPLEEXAMPLE55

44Quick Check

262

2z2

EXAMPLEEXAMPLE44

33Quick Check

2521

2t21

EXAMPLEEXAMPLE33

12 Solving Inequalities With Variables on Both Sides

p R –1 m K –3 b S 3

Test-Taking Tip

1 A B C D E

2 A B C D E

3 A B C D E

4 A B C D E

5 A B C D E

B C D E

To grid a mixednumber, write it as animproper fraction:

.315 5 16

5

9 9 9 9

0 0 01 1 1 12 2 2 23 3 3 34 4 4 45 5 5 56 6 6 67 7 7 78 8 8 8

. . . ./ /1 6 / 5

nline

Visit: PHSchool.comWeb Code: ate-0775

221

Alternative Method

Work the example a second wayon the board by subtracting 6zfrom both sides. Lead students tosee that division by a negativenumber is unnecessary if youchoose the operation that resultsin a positive coefficient. Then, youdo not have to remember toreverse the inequality sign.

Additional Examples

Solve 8z - 6 � 3z + 12. z R 3

Solve 5(-3 + d) # 3(3d - 2). d L –2


Adapted Instruction

Closure

Instruct students to write a multi-step inequality with variables onboth sides that requires the use ofthe distributive property. Havestudents exchange inequalitiesand solve.

L1

L3

14

55

35

44

EXAMPLEEXAMPLE44

PowerPoint


EXERCISES For more

practice, see Extra Practice.Practice and Problem Solving

Solve each inequality. Check your solution.

1. 4d 1 7 # 23 d K 4 2. 5m 2 3 . 218 m S –3 3. 24x 2 2 , 8

4. 5 2 3n $ 24 n K 3 5. 8 # 212 1 5q q L 4 6. 5 # 11 1 3h h L –2

7. 27 # 5 2 4a a K 3 8. 10 . 29 2 3b b S 6 9. 5 2 9c . 213 c R 2

Write and solve an inequality.

10. On a trip from Virginia to Florida, the Sampson family wants to travel at least420 miles in 8 hours of driving. What must be their average rate of speed?

11. Geometry The perimeter of an isosceles triangle is at most 27 cm. One side is8 cm long. Find the possible lengths of the two congruent sides.

12. You want to solve an inequality containing the expression 23(2x 2 3). Thenext line in your solution would rewrite this expression as 9.–6x ± 9


13. 2(j 2 4) $ 26 j L 1 14. 2(6b 2 2) . 0 b R 15. 22(h 1 2) , 214

16. 23 # 3(5x 2 16) x L 3 17. 25 . 2(4y 1 7) y S –818. 4(w 2 2) # 10

19. 23(c 1 4) 2 2 . 7 20. 22(r 2 3) 1 7 $ 8 21. 16 # 4 2 3(n 2 13)

22. 3w 1 2 , 2w 1 5w R 323. 3t 1 7 $ 5t 1 9 t K –1 24. 4d 1 7 $ 1 1 5d

25. 5 2 2n # 3 2 n n L 2 26. 2k 2 3 # 5k 1 9 k L –427. 3s 1 16 . 6 1 4s

28. 6p 2 1 . 3p 1 8 p S 3 29. 3x 1 2 . 24x 1 16 30. 2 2 3m , 4 1 5m

31. 23(v 2 3) $ 5 2 4v 32. 3q 1 6 # 25(q 1 2) 33. 3(2 1 r) $ 15 2 2r

34. 9 1 x , 7 2 2(x 2 3) 35. 2(m 2 8) , 28 1 3m 36. 2v 2 4 # 2(3v 2 6)

Tell what you must do to the first inequality in order to get the second.

37. 8 2 4s . 16; 24s . 8 38. g 1 7 $ 9; g $ 2

39. 2y 2 5 . 9 1 y; y . 14 See left. 40. 28 . 2 2; 30 , z See left.

41. Writing Suppose a friend is having difficulty solving 2.5(p 2 4) . 3(p 1 2).Explain how to solve the inequality, showing all necessary steps and identifyingthe properties you would use. See margin.

42. Multiple Choice Mandela is starting a part-time word-processing business out of hishome. He plans to charge $15 perhour. The table at the right showshis expected monthly businessexpenses. Which inequalitydescribes the number of hours h hemust work in a month to make aprofit of at least $600? C

15h 1 600 # 490 1 45 1 6515h $ 490 1 45 1 65 1 60015h 1 600 $ 490 1 45 1 6515h $ 600

Equipment rentalMaterialsBusiness phone

$490$45$65

Expense Cost

z25

23

23

Apply Your SkillsBB

Example 5(page 221)

Example 4(page 221)

13

Example 3(page 221)

Example 2(page 220)

13

Example 1(page 220)

27 L 2s ± 8 and s K 9.5, so the two equal sides must be no longer than 9.5 cm.

Subtract 8from each side.

Subtract 7 fromeach side.

h S 5w K 41

2

n K 9

d K 6

s R 10

x S 2

v L –4

x R 113

q K –2

m S –8

r L 145

v L 2

m S –14

c R –7 r K 212

39. Add 5 and subtracty from each side.

40. Add 2 to each side,then multiply eachside by –5 andreverse theinequality sign.



8t L 420 and t L 52.5, so the average rate of speed must be at least 52.5 mi/h.

x S –212

GO forHelp

222


41. Answers may vary.Sample: To solve 2.5(p – 4) S 3(p ± 2),first use the DistributiveProperty to simplify both

sides. The result is 2.5p – 10 S 3p ± 6. Thenuse the SubtractionProperty of Inequality.Subtract 6 from each sideand 2.5p from each side.The result is –16 S 0.5p.

Then use the DivisionProperty of Inequality.Divide each side by 0.5.The result is –32 S p. So the solution is –32 S p or p R –32.


A B 1-12, 42-50, 70-71

A B 41, 51-69, 72-73C Challenge 74-79



Error Prevention!

Exercises 1–9 Simple mistakes canbe made while checking asolution, thus making you thinkyour answer is incorrect when it iscorrect. Remind students that it iseasiest to check an answer bysubstituting 0 for the variable ineach solution.

2

1


Enrichment

Reteaching

Adapted Practice

Name Class Date

© P

ears

on E

duc

atio

n, In

c. A

ll rig

hts

rese

rved

.

Practice 4-4 Percent of Change

Find each percent of change. Describe the percent of change as an increaseor decrease. Round to the nearest whole number.

1. 36 g to 27 g 2. 40 cm to 100 cm 3. 90 in. to 45 in. 4. 500 lb to 1500 lb

5. $90 to $84.50 6. $100 to $140 7. $15 to $5.50 8. 100 mi to 175 mi

9. 280 m to 320 m 10. 58 to 76 11. 60 to 150 12. 600 mi to 480 mi

13. 18 to 27 14. 290 yd to 261 yd 15. 26.2 to 22.8 16. $8.50 to $12.75

17. to 18. to 19. to 20. to

Find each percent of change. Describe the percent of change as an increaseor decrease. Round to the nearest whole number.

21. In 1985, the average price for gasoline was $1.20/gal. In 2000, theaverage price for gasoline was $1.56. Find the percent of change.

22. In 1980, Texas had 27 U.S. Representatives. That number increased to 30 in 2000. Find the percent of change.

23. In 1980, the average annual tuition charge for a four-year publicuniversity was $840. The average annual tuition charge in 2000 was$3356. What is the percent of change?

24. The United States imported 6,909,000 barrels of oil per day in 1980. In2000, the United States imported 11,459,000 barrels of oil per day. Whatis the percent of change?

25. In 1977, the average number of households with cable television was16.6%. In 2000, the average number of households with cable televisionwas 68%. What is the percent of change?

26. In 1989, there were 38,000 licensed drivers under the age of 16. In 1999,the total number of licensed drivers under 16 was 33,248. Find thepercent of change.

27. In 1990, Atlanta, GA, failed to meet air quality standards on 42 days.In 1999, Atlanta failed to meet air quality standards on 61 days. What isthe percent of change?

Find the greatest possible error and the percent error for each measurement.

28. 3 cm 29. 0.5 cm 30. 6 cm

31. 16 in. 32. 36.85 g 33. 0.9 cm

Find the minimum and maximum possible areas for rectangles with thefollowing measurements.

34. 8 cm 3 10 cm 35. 3 in. 3 5 in. 36. 8 m 3 12 m

Find the minimum and maximum possible volume for a rectangular solidwith the following measurements.

37. 16 in. 3 22 in. 3 18 in. 38. 13 cm 3 15 cm 3 18 cm 39. 3 m 3 4 m 3 5 m

1814151

281463

46612743

42914361

2

Practice

L3

L4

L2

L1

L3


43. Expenses The sophomore class is planning a picnic. The cost of a permit to usea city park is $250. To pay for the permit, there is a fee of $.75 for eachsophomore and $1.25 for each guest who is not a sophomore. Two hundredsophomores plan to attend. Write and solve an inequality to find how manyguests must attend for the sophomores to pay for the permit. See left.

44. Health Care Systolic blood pressure is the higher number in a blood pressure reading. It is measured as your heart muscle contracts. The formulaP # a 1 110 gives the normal systolic blood pressure P based on age a.a. At age 20, does 120 represent a maximum or a minimum normal

systolic pressure? maximumb. Find the normal systolic blood pressure for a 50-year-old person.

Match each inequality with its graph below.

45. 22x 2 2 . 4 B 46. 2 2 2x . 4 E 47. 2x 1 2 . 4 F

48. 2x 1 2 . 4x A 49. 2x 2 2 . 4 D 50. 22(x 2 2) . 4 C

A. B.

C. D.

E. F.

51. Open-Ended Write two different inequalities that you can solve by adding 5and multiplying by 23. Solve each inequality. Answers may vary.


52. r 2 3 , r 1 2 r r R 5 53. 4 2 2m # 5 2 m 1 1 m L –2

54. 22(0.5 2 4s) $ 23(4 2 3.5s) s K 4.4 55. n 2 $ 1 n n K –2

56. 2(8 2 s) , 0 s R 8 57. 3.8 2 k # 5.2 2 2k k K 1.4

58. 10 . 3(2n 2 1) 2 5(4n 1 3) n S –2 59. 3(3r 1 1) 2 (r 1 4) # 13 r K 1

60. 2(3x 1 7) . 4(7 2 2x) x S 1 61. 4(a 2 2) 2 6a # 29 a L

62. 4(3m 2 1) $ 2(m 1 3) m L 1 63. 17 2 (4k 2 2) $ 2(k 1 3) k K 2

64. 2n 2 3(n 1 3) # 14 n L–23 65. 5x 2 (3x 1 8) # 24 1 3x x K 0

66. 5a 2 2(a 2 15) , 10 a R–6 67. 5c 1 4(c 2 1) $ 2 1 5(2 1 c)

68. a. Solve 5t 1 4 # 8t 2 5 by gathering the variable terms on the left side andthe constant terms on the right side of the inequality.–3t K –9, t L 3

b. Solve 5t 1 4 # 8t 2 5 by gathering the constant terms on the left side andthe variable terms on the right side of the inequality. 9 K 3t, t L 3

c. Compare the results of parts (a) and (b). The results are the same.

69. a. Mental Math Like equations, some inequalities are true for all values of thevariable, and some inequalities are not true for any values of the variable.Determine whether each inequality is always true or never true.i. 4s 1 6 $ 6 1 4s ii. 3r 1 5 . 3r 2 2 iii. 4(n 1 1) , 4n 2 3

b. Critical Thinking How can you tell whether an inequality is always true ornever true without solving? See margin.

23

12

16

12

34

58

56

34

18

12

12

13

23

43

�1�2�3 0 1 2 3 4 5�1�2�3�4�5 0 1 2 3

�1�2�3 0 1 2 3 4 5�1�2�3�4�5 0 1 2 3

�1�2�3�4�5 0 1 2 3�1�2�3�4�5 0 1 2 3

12


Normal blood pressure forteens is about 110/70.

Samples: – x – 5 S 0, x R –15; – x – 5 K 10, x L –451

3

13

no more than 135

43. For x ≠ the numberof guests, (0.75)200± 1.25x L 250, x L80, so at least 80guests must attend.

c L 4

Inequalities or equationsthat are always true arecalled identities. (See p. 136)

Vocabulary Tip

GPS

See margin.GO nlineHomework HelpVisit: PHSchool.comWeb Code: ate-0404

223

Connection to GeometryExercise 11 Some students maywant to use the formula for theperimeter of a rectangle. Tellthem there is no set formula forthe perimeter of a triangle. Ask:What is special about an isoscelestriangle? Two sides are equal.Instruct students to keep this inmind when they are writing theirinequality.

Exercise 44 Normal systolic bloodpressure for an adult is from 90 to 140.

69a. i. always trueii. always trueiii. never true

b. If the coefficients of theonly variable on each sideof an inequality are thesame, then the inequalitywill either be always trueor never true.


70. Commission Joleen is a sales associate in a clothing store. Each week she earns$250 plus a commission equal to 3% of her sales. This week her goal is to earnno less than $460. Write and solve an inequality to find the dollar amount of thesales she must have to reach her goal. See left.

71. A student uses the table below to help solve .

a. Critical Thinking Based on the table, would you expect the solution ofto be of the form or ? Explain.

b. Estimate Based on the table, estimate the value of n.c. Solve the inequality. Compare the actual solution to your estimated solution.

Error Analysis Find and correct the mistake in each student’s work.

72. 73.

74. a. Solve ax 1 b . c for x, where a is positive. x Sb. Reasoning Solve ax 1 b . c for x, where a is negative. x R

75. Geometry The base of a triangle is 10 in. Its height is (x 1 4) in. Its area is nomore than 56 in.2. What are the possible integer values of x?

76. Architecture The rectangle shown on the building at the left is a goldenrectangle. Artists often use the golden rectangle because they consider it to bepleasing to the eye. The ratio of two sides of a golden rectangle isapproximately 1 : 1.62. Suppose you are making a picture frame in the shape ofa golden rectangle. You have a 46-in. length of wood to use for a frame. Whatare the dimensions of the largest frame you can make? Round to the nearesttenth of an inch. 8.8 in. � 14.2 in.

77. Critical Thinking Find a value of a such that the number line below shows allthe solutions of ax 1 4 # 212.–2

78. Earning You can earn money by handing out flyers in the afternoon for $6.50an hour and by typing a newsletter in the evening for $8 an hour. You have 20hours available to work. What are the greatest number of hours you can spendhanding out flyers and still make at least $145? 10 h

79. Freight Handling The freight elevator of a building can safely carry a load of atmost 4000 lb. A worker needs to move supplies in 50-lb boxes from the loadingdock to the fourth floor of the building. The worker weighs 160 lb. The cart sheuses weighs 95 lb.a. What is the greatest number of boxes she can move in one trip? 74 boxesb. The worker must deliver 310 boxes to the fourth floor. How many trips must

she make? 5 trips

4 5 6 7 8 9�1 3210

c 2 ba

c 2 baChallengeCC

� 3n � 1 4(n � 2)� 3n � 1 4n � 2� 3n � 1 4n� � 1 n

� �2x � 5 3x � 3� �2x � 2 3x� 2 x

x . nx , n6x 1 1 , 5(3 2 x)

6x 1 1 , 5(3 2 x)


The Parthenon, an ancientGreek temple, has dimensionsthat form a golden rectangle.

1.62x

x

x is an integer between –3 and 7, inclusive.

See back of book.

70. For x ≠ amount ofsales, 250± 0.03xL460, so to reach hergoal, she must haveat least $7000 insales.

71a. x R n; the greatervalues in the tablemake the inequalityfalse.

b. Sample: 1.3c. x R 1 ; this is close

to the estimate.

311

x

0

6x � 1

6(0) � 1 � 1

0.5 6(0.5) � 1 � 4

1 6(1) � 1 � 7

1.5 6(1.5) � 1 � 10

<

true

true

true

false

5(3 � x)

5(3 � 0) � 15

5(3 � 0.5) � 12.5

5(3 � 1) � 10

5(3 � 1.5) � 7.5

For a guide to solvingExercise 72, see p. 226.

for HelpGO

224

Lesson Quiz


1. 8 + 5a $ 23 a L 3

2. - p � p - 6 p S 7

3. 3(x - 4) � 4x + 7 x R –19

4. 3(3c + 2) # 2(3c - 2) c K –3


Have each student graph x � -3on a number line. Instruct eachstudent to work backwards fromthis inequality to write aninequality that can be solvedusing at least three steps.

13

15

12

13

PowerPoint

4. Assess & Reteach



80. The Science Club hopes to collect at least 200 kg of aluminum cans for recycling thissemester (21 weeks). The graph at the rightshows the first week’s results.

Let x represent the average mass of cansrequired per week for the remainder of thesemester. Which inequality would you use tofind x? C

A. x $ B. x $

C. x $ D. x . 2 8

81. Solve 2x 2 8 . 4x 1 2. FF. x , 25 G. x . 25 H. x , 5 J. x . 5

82. Solve 25n 1 16 # 27n. AA. n # 28 B. n $ 28 C. n # 8 D. n $ 8

83. Great Gifts pays its supplier $65 for each box of 12 bells. The owner wants to determine the least amount x he can charge his customers per bell in order to make at least a 50% profit per box. Which inequalityshould he use? FF. 12x $ 1.50(65) G. 65x # 1.50(12)H. 0.50(12x) $ 65 J. 0.50(12x) # 65

84. Maxwell orders at least 30 bottles of flea shampoo per month for his pet-grooming business. His supplier charges $3 per quart bottle plus a $25handling fee per order. A competing supplier offers a similar product for $4per quart bottle plus a $5 handling fee per order. The salesman for thecompetitor shows Maxwell that 10 bottles from his company would costonly $45 compared to $55 from Maxwell’s current supplier.

Which supplier would you advise Maxwell to use? Explain or show work tosupport your advice to Maxwell. See margin.

Mixed Review


85. 29m $ 36 86. 224 # 3yy L –8 87. . 24 x S –12

88. # 1 89. b , 18 b R 27 90. 42 . w

91. 56 , 42p p S 1 92. 0.5d $ 3.5 93. . 10

94. Your family leaves your town traveling at an average rate of 45 mi/h. Two hourslater, your neighbor leaves your town along the same road at an average rate of60 mi/h. How many hours will it take your neighbor to overtake you? 6 h

Simplify each expression.

95. 242 –16 96. (24)2 16 97. (22)3(23) 24 98. 224

–16

Lesson 2-3

Lesson 3-6

x5

13

237

232t

3

x3

Lesson 4-3

Short Response

Q20020 R

(20028)20

(200 2 8)21

20021

Mon.Day

Wed. Fri.

5

4

3

2

1

0

Mas

s (k

g)

Aluminum CansCollected in Week 1

Multiple Choice

Test Prep

m K –4

t L –3

d L 7

w S –98

x S 50



GO forHelp

225

Test Prep


Transparencies

Exercise 80 Remind students tolook for key words and phrases,such as remainder of thesemester.


84. [2] Maxwell shouldcontinue ordering fromthe current supplier.

Let x ≠ number ofbottles Maxwell ordersper month. The costfrom the currentsupplier is 3x ± 25. The cost fromthe competitor is 4x ±

5. The inequality 4x ± 5 K 3x ± 25represents the numberof bottles of shampoofor which thecompetitor will be lessexpensive. Since x ≠

20, the competitor willnot be less expensivefor orders of 30 bottlesor more. (ORequivalent explanation)

[1] incorrect answer ORinsufficient explanation

Part 1

Two inequalities that are joined by the word and or the word or form a

You can write the compound inequality x $-5 and x # 7 as -5 # x # 7.

The graph above shows that a solution of -5 # x # 7 is in the overlap of thesolutions of the inequality x $ -5 and the inequality x # 7.

You can read -5 # x # 7 as “x is greater than or equal to -5 and less than orequal to 7.” Another way to read it is “x is between -5 and 7, inclusive.”

Writing a Compound Inequality

Write a compound inequality that represents each situation. Graph the solutions.

a. all real numbers that are at b. Today’s temperatures will beleast -2 and at most 4 above 32°F, but not as high as 40°F.n $ -2 and n # 4 32 , t and t , 40-2 # n # 4 32 , t , 40

Write a compound inequality that represents each situation. Graph your solution.a. all real numbers greater than -2 but less than 9 b. The books were priced between $3.50 and $6.00, inclusive. 3.50 K b K 6

11Quick Check3433323130 35 36 37 38 39 40 41 42�2 �1 0 1 2 3 4�3 5

EXAMPLEEXAMPLE11

1 3 4 5 6 7 820�3�4�5�6 �2 �1�5 � x � 7

x � �5 x � 7

compound inequality.

Compound Inequalities

Lesson 4-5 Compound Inequalities 227

4-54-5

Lessons 1-1 and 4-1

Graph each pair of inequalities on one number line. 1–3. See below.

1. c , 8; c $ 10 2. t$ –2; t# –5 3. m# 7; m. 12

Use the given value of the variable to evaluate each expression.

4. 3n - 6; 4 6 5. 7 - 2b; 5 –3

6. ; 17 14 7. ; 9 3

New Vocabulary • compound inequality

2d 2 35

12 1 13 1 y3

What You’ll Learn• To solve and graph

inequalities containing and

• To solve and graphinequalities containing or

. . . And WhyTo solve a problem involvingthe chemistry of a swimmingpool, as in Example 3

11 Solving Compound Inequalities Containing And

1.

2.�5�6 �4�3�2�1

7 8 96 1210 11

See graphs below.

a. 0 2 4�4�2 106 8

b. 2 3 40 1 75 6

1a. n S –2 and n R 9, or–2 R n R 9

3.7 8 95 6 12 13 1410 11


The word inclusive isrelated to the wordincluded.

Vocabulary Tip

See above left.

4-54-5

227

1. PlanObjectives1 To solve and graph

inequalities containing and2 To solve and graph

inequalities containing or

Examples1 Writing a Compound

Inequality2 Solving a Compound

Inequality Containing And3 Real-World Problem Solving4 Writing Compound

Inequalities5 Solving a Compound

Inequality Containing Or

Math Background

In logic, and means that a number or variable satisfies bothconditions given. The word ormeans that only one conditionmust be satisfied. Mathematicsuses these meanings from formal logic.

More Math Background: p. 198D





Using Variables Lesson 1-1: Example 2Extra Skills and Word


Inequalities and Their GraphsLesson 4-1: Example 3Extra Skills and Word


PowerPoint

Special NeedsIllustrate the two conjunctions by asking studentswearing blue to stand (and then sit). Then askstudents wearing yellow to do the same, and thenthose wearing blue and yellow, and finally, thosewearing blue or yellow.

Below LevelHave three students stand on a number line on thefloor. Discuss how the middle student is “greaterthan” one student and “less than” the other at thesame time.

L2L1

learning style: tactile learning style: tactile


A solution of a compound inequality joined by and is any number that makesboth inequalities true. One way you can solve a compound inequality is bywriting two inequalities.

Solving a Compound Inequality Containing And

Solve -4 , r - 5 # -1. Graph your solution.

Write the compound inequality as two inequalities joined by and.

-4 , r - 5 and r - 5 # -1

-4 + 5 , r - 5 + 5 P r - 5 + 5 # -1 + 5 Solve each inequality.

1 , r and r # 4 Simplify.

1 , r # 4

Solve each inequality. Graph your solution.a. -6 # 3x , 15 b. -3 , 2x - 1, 7 c. 7 ,-3n+ 1 # 13

You could also solve an inequality like -4 , r - 5 # -1 by working on all threeparts of the inequality at the same time. You work to get the variable alonebetween the inequality symbols.

Chemistry The acidity of the water in a swimming pool is considered normal if theaverage of three pH readings is between 7.2 and 7.8, inclusive. The first tworeadings for a swimming pool are 7.4 and 7.9. What possible values for the thirdreading p will make the average pH normal?

Relate 7.2 7.8

Write 7.2 7.8

7.2 # # 7.8

3(7.2) # 3 # 3(7.8) Multiply by 3.

21.6 # 15.3 + p # 23.4 Simplify.

21.6 - 15.3 # 15.3 + p - 15.3 # 23.4 - 15.3 Subtract 15.3.

6.3 # p # 8.1 Simplify.

The value for the third reading must be between 6.3 and 8.1, inclusive.

a. Suppose the first two readings for the acidity of water in a swimming pool are7.0 and 7.9. What possible values for the third reading will make the average pH normal? 6.7 K p K 8.5

b. Critical Thinking If two readings are 8.0 and 8.4, what possible values for thethird reading will make the average pH normal? Are these third readings likely? Explain.

33Quick Check

Q7.4 1 7.9 1 p3 R

7.4 1 7.9 1 p3

#7.4 1 7.9 1 p

3#

which is less thanor equal to

the averageis less thanor equal to


22Quick Check

0 1 2 3 4 5�5 �4 �3 �2 �1

EXAMPLEEXAMPLE22


The lifeguard is checking thepH of swimming pool water.The pH of a substance is ameasure of how acidic or basicit is. pH is measured on a scalefrom 0 to 14. Pure water isneutral, with a pH of 7.

5.2 K p K 7. No; readings in this range are unlikely if thefirst readings are high.

–2 K x R 5; –1 R x R 4;

c. –4 K n R –2;

�4�3�2�5 1�1 0

�4 0�2 42 6 0 1 2�2�1 53 4

228

2. Teach

Guided Instruction

Alternative Method

Graph the example on the boardusing colored chalk. Use yellowchalk for one inequality and bluechalk for the other inequality.Shade where the two graphsoverlap. Help students see thatthe solution for an inequalitycontaining and is the area wherethe graphs overlap.

Additional Examples

Write a compound inequalitythat represents each situation.Graph the solutions.a. all real numbers that are atleast -1 and at most 3 b L –1 and b K 3–1 K b K 3

b. all real numbers that are lessthan 31, but greater than 25 n R 31 and n S 2525 R n R 31

Solve 5 � 5 - ƒ � 2. Graphyour solution. 0 R ƒ R 3

Your test grades in science sofar are 83 and 87. What possiblegrades can you make on your nexttest to have an average between85 and 90, inclusive? The thirdtest grade must be between 85and 100, inclusive.

33

�5�4�3�2�1 0 54321

22

23 24 25 26 27 28 3332313029

�5�4�3�2�1 0 54321

11

EXAMPLEEXAMPLE22

Advanced LearnersAsk students to explain how to write compoundinequalities using or or and where solutions are allreal numbers.

English Language Learners ELLTo be sure that students understand the concept, askthem to write their own compound inequalities usingand as well as or. Ask volunteers to present andexplain their examples to the class.

L4


PowerPoint

Part 2

A solution of a compound inequality joined by or is any number that makes eitherinequality true.

Writing Compound Inequalities

Write a compound inequality that represents each situation. Graph the solution.

a. all real numbers that are less than -3 or greater than 7x , -3 or x . 7

b. Discounted fares are available to children 12 and under or to adults at least60 years of age.n # 12 or n $ 60; n $ 0 because age cannot be negative.

Write an inequality that represents all real numbers that are at most -5 or at least 3. Graph your solution. n K –5 or n L 3;

For a compound inequality joined by or, you must solve each of the twoinequalities separately.

Solving a Compound Inequality Containing Or

Solve the compound inequality 4v + 3 ,-5 or -2v + 7 , 1. Graph the solution.

4v + 3 , -5 or -2v + 7 , 1

4v + 3 - 3 , -5 - 3 -2v + 7 - 7 , 1 - 7

4v , -8 ∞ -2v , -6

, .

v , -2 or v . 3

Solve the compound inequality -2x + 7 . 3 or 3x - 4 $ 5. Graph your solution.

EXERCISES For more practice, see Extra Practice..


Write a compound inequality that represents each situation. Graph your solution.

1. all real numbers that are between -4 and 6

2. all real numbers that are at least 2 and at most 9

3. The circumference of a baseball is between 23 cm and 23.5 cm.

4. Tropical Storm The wind speeds of a tropical storm are at least 40 mi/h but nomore than 74 mi/h. 40 K w K 74;

30 5040 7060 80

Example 1(page 227)

55Quick Check0 1 2 3 4 5�5 �4 �3 �2 �1

2622

22v22

284

4v4

EXAMPLEEXAMPLE55

�4�2 0�6 62 4

44Quick Check30 40 50 60 7020100

�2 �1 0 1 2 3 4�3 5�4 6 7 8

EXAMPLEEXAMPLE44

12 Solving Compound Inequalities Joined by Or



–4 R x and x R 6, or –4 R x R 6;

2 K n and n K 9, or 2 K n K 9;

23 R c R 23.5;

x R 2 or x L 3; �1 10 32 4


�2 0 2�6�4 84 6

0 2 4�2 106 8

22 23 24

For: Compound InequalityActivity

Use: Interactive Textbook, 4-5

GO forHelp

229

Additional Examples

Write an inequality thatrepresents each situation. Graphthe solution.a. all real numbers that are lessthan 0 or greater than 3 n R 0 orn S 3

b. Discounted tickets areavailable to children under 7 yearsold or to adults 65 and older. a R 7 or a L 65; because agecannot be negative, a L 0

Solve the compound inequality3x + 2 � -7 or -4x + 5 � 1.Graph the solution. x R –3 or x S 1


Adapted Instruction

Closure

Ask students to explain in theirown words the differencebetween compound inequalitieswith and and those with or.Compound inequalities with andhave solutions that satisfy everypart of the inequality. Compoundinequalities with or have solutionsthat satisfy at least one part ofthe inequality.

L1

L3

�5�4�3�2�1 0 54321

55

0 10 10090807050 60403020

�5�4�3�2�1 0 54321

44

PowerPoint


Solve each compound inequality. Graph your solution.

5. -3 , j + 2 , 7 6. 3 # w + 2 # 7 7. 2 , 3n - 4 # 14

8. 7 # 3 - 2p , 11 9. -2 , -3x + 7 , 4 10. 1.5 , w + 3 # 6.5

11. -16 , -3x + 8 , -7 12. -1 , 4m + 7 # 11 13. -9 , -2s - 1 # -7

14. 12 # # 16 15. , , 5 16. -2 # # 2

For each situation write and graph an inequality.

17. all real numbers n that are at most -3 or at least 5

18. all real numbers x that are less than 3 or greater than 7

19. all real numbers h less than 1 or greater than 3

20. all real numbers b less than 100 or greater than 300

Solve each compound inequality. Graph your solution.

21. 3b - 1 , -7 or 4b + 1 . 9 22. 4 + k . 3 or 6k , -30

23. 3c + 4 $ 13 or 6c - 1 , 11 24. 6 - a , 1 or 3a # 12

25. 7 - 3c $ 1 or 5c + 2 $ 17 26. 5y + 7 # -3 or 3y - 2 $ 13

27. 2d + 5 # -1 or -2d + 5 # 5 28. 5z - 3 . 7 or 4z - 6 , -10

Write a compound inequality that each graph could represent.

29. –2 R x R 3

30. x R –3 or x L 2

31. x K 0 or x S 2

32. –4 K x K 3

Solve each compound inequality.

33. 3q - 2 . 10 or 3q - 2 # -10 34. 3 - 2h . 17 or 5h - 3 . 17

35. 1 # 0.25t # 3.5 36. 25r , 400 or 100 , 4r

37. -20 # 3t - 2 , 1 38. - 4 . 3 or . 3

39. Multiple Choice The force exerted on a spring is proportional to thedistance the spring stretches from its relaxed position. Suppose you stretcha spring distance d in inches by applying force F in pounds. For a certainspring, = 0.8. You apply forces between 25 and 40 pounds, inclusive.Which inequality describes the stretch of the spring? D

40. Reasoning Describe the solutions of 3x - 8 , 7 or 2x - 9 . 1.

41. Writing Explain the difference between the words and and or in a compound inequality.

20 # d # 3231.25 # d # 4020 , d , 3225 # d # 40

dF

3 2 2x5

3x 1 14

0 1 2 3 4�4 �3 �2 �1

0 1 2 3 4�4 �3 �2 �1

0 1 2 3 4�4 �3 �2 �1

0 1 2 3 4�4 �3 �2 �1

Apply Your SkillsBB

Example 5(page 229)

3000 100

0 21 43 5

Example 4(page 229)

5 2 x3

3x 2 14

12

14 1 17 1 a3

Examples 2, 3(page 228)

at rest

d in.

q K –2 or q S 423

4 K t K 14

–6 K t R 1

h R –7 or h S 4

r R 16 or r S 25

x R –6 or x S 9

all real numbers except 5

The word and means both statements must be true.The word or means that at least one of the statements must be true.

–5 R j R 5 1 K w K 5 2 R n K 6

–4 R p K –2

5 R x R 8 –2 R m K 1 3 K s R 4

5 K a K 17

n K –3 or n L 5

b R –2 or b S 2

c R 2 or c L 3

c K 2 or c L 3

d K –3 or d L 0

k R –5 or k S –1

a K 4 or a S 5

y K –2 or y L 5

z R –1 or z S 2

x R 3 or x S 7

h R 1 or h S 3

b R 100 or b S 300

1 R x R 7 –1 K x K 11

1 R x R 3 –1.5 R w K 3.5



�2 0 2�4 4 6

0 2 4�2 6 8


5.

6.1 2 3�1 0 64 5

�10 �5 0 5 10

7.

8.

9.�1 10 32 4

�4 �3�2�5 1�1 0

�2 20 64 810.

11.

12.�3�2�1 0 1 2

4 65 87 9

�1�2 0 1 2 3 4


A B 1-16, 35-39, 42-49

A B 17-34, 40-41C Challenge 50-55


Homework Quick CheckTo check students’ understandingof key skills and concepts, go overExercises 12, 20, 39, 41, 46-48.

Connection To GeometryExercises 42–45 Point out thatthe length of the third side mustbe between the sum and thedifference of the two given sides.Ask: Can the length of the thirdside be 4 cm? no Why or Whynot? The sum of 3 and 7 is 10;the difference is 4. The length 4 isnot between 10 and 4.

Error Prevention!

Exercises 46–48 Students mayhave difficulty reading the graph.Ask questions about the high andlow temperatures for each city tohelp students understand thedifferent bars.

2

1


Enrichment

Reteaching

Adapted Practice

Name Class Date

© P

ears

on E

duc

atio

n, In

c. A

ll rig

hts

rese

rved

.

Practice 4-5 Applying Ratios to Probability

A driver collected data on how long it takes to drive to work.

1. Find P(the trip will take 25 min).

2. Find P(the trip will take 20 min).

3. Find P(the trip will take at least 25 min).

Use the data in the line plot to find each probability.

4. P(June) 5. P(October) 6. P(first six months of year)

7. P(May) 8. P(not December) 9. P(last three months of year)

A cereal manufacturer selects 100 boxes of cereal at random. Ninety-nine ofthe boxes are the correct weight. Find each probability.

10. P(the cereal box is the correct weight)

11. P(the cereal box is not the correct weight)

12. There are 24,000 boxes of cereal. Predict how many of the boxes are the correct weight.

13. One letter is chosen at random from the word ALGEBRA. Findeach probability.

a. P(the letter is A) b. P(the letter is a vowel)

14. Patrice has a 40% chance of making a free throw. What is theprobability that she will miss the free throw?

15. A box of animal crackers contains five hippos, two lions, three zebras, and four elephants. Find the probability if one animal cracker is chosen at random.

a. P(a hippo) b. P(not an elephant)

c. P(an elephant or a lion)

16. Anthony is making a collage for his art class by picking shapesrandomly. He has five squares, two triangles, two ovals, and four circles. Find each probability.

a. P(circle is chosen first) b. P(a square is not chosen first)

c. P(a triangle or a square is chosen first)

Time in minutes 20 25 30

Number of trips 4 8 2

JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC

X

X

X

X X

X

X

X X

X

X

X

X

X

X X

X

X

X

X

X X

X

X

X

Student Birth Months

Practice

L3

L4

L2

L1

L3

230

231

13.

14.

15.0 2 4�2 106 8

10 150 5 20

2 3 4 516.

21.

22.�4 �3�2�6�5 1�1 0

�3 �2 �1 0 1 2 3

2 4 6�2 0 128 1023.

24.

25.0 21 43 5

2 43 65 7

�1 10 32 4

Lesson Quiz

1. Write two compoundinequalities that represent thegiven situation. Graph thesolution.

all real numbers that are atleast 2 and at most 5 b L 2 and b K 5, 2 K b K 5

2. Write an inequality thatrepresents the given situation.Graph the solution.

all real numbers that are lessthan -3 or greater than -1 nR –3 or n S –1

3. Solve -2 # 2x - 4 � 6. Graphthe solution. 1 K x R 5

4. Solve 3x - 2 � -8 or -2x + 5 # 3. Graph thesolution. x R –2 or x L 1


Organize students in groups offour and instruct them to sit in acircle. Instruct each student towrite a real world problem similarto Example 3 on a piece of paper.Have students pass their problemsto the student on their right. Thisstudent writes an inequality torepresent the problem. Pass theproblems to the right again. Thisstudent solves the inequality. Passthe problems to the right again.This student graphs the solutions.

�5�4�3�2�1 0 54321

�3�2�1 0 5 6 74321

�5�4�3�2�1 0 54321

�3�2�1 0 5 6 74321

PowerPoint

4. Assess & Reteach


Geometry The sum of the lengths of any two sides of a triangle is greater than thelength of the third side. The lengths of two sides of a triangle are given. Find therange of values for the possible lengths of the third side.

Sample 3 cm, 7 cm

Write inequalities for x as the longest side and for7 cm as the longest side. The length 3 cm cannot bethe longest side.

x + 3 . 7 and 3 + 7 . x

x . 4 and 10 . x Solve each inequality.

4 , x , 10

The length of the third side is greater than 4 cm and less than 10 cm.

42. 2.5 in., 5 in. 43. 12 ft, 18 ft 44. 28 mm, 21 mm 45. 5 m, 16 m

Meteorology The graph below shows the average monthly high and lowtemperatures for Detroit, Michigan, and Charlotte, North Carolina.

46. Write a compound inequality for Charlotte’saverage temperature in June. 66 K C K 88

47. Write a compoundinequality for Detroit’saverage temperature inJanuary. 15 K D K 30

48. Write a compoundinequality for the yearlytemperature range foreach city.

49. Open-Ended Describe a real-life situation that you could represent with theinequality -2 , x , 8.

50. Nursing In nursing school, students learn temperature ranges for bath water.Tepid water is approximately 80°F to 93°F, warm water is approximately 94°Fto 98°F, and hot water is approximately 110°F to 115°F. Model these ranges onone number line. Label each interval.

Write a compound inequality that each graph could represent.

51. 52.

53. Pulse Rates When you exercise, your pulse rate rises. Recommended pulserates vary with age and physical condition. For vigorous exercise, such asjogging, the inequality 0.7(220 - a) # R # 0.85(220 - a) gives a target rangefor pulse rate R (in beats per minute), based on age a (in years).a. What is the target range for pulse rates for a person 35 years old? Round to

the nearest whole number.b. Your cousin’s target pulse rate is in the range between 140 and 170 beats per

minute. What is your cousin’s age? 20 years old

54. Find three consecutive even integers whose sum is between 48 and 60. 16, 18, 20

55. Find three consecutive even integers such that one half of their sum is between15 and 21. 10, 12, 14

�2 0 2 4�2 0 2 4�4

ChallengeCC

SOURCE: Statistical Abstract of the United States

J F M A MMonthJ J A S O N D

0

20

40

60

80

100

Tem

pera

ture

(º F

)

Monthly Average High and Low Temperatures

Detroit Charlotte

3 cm 7 cm

x


To estimate your pulse rate,count the number of beatsyou feel in 15 seconds at apressure point. Multiply thisnumber by 4.

Charlotte: 29 K C K 90Detroit: 15 K D K 83

–2 R x R 0 or 0 R x R 3 K 0 or n L 3un u

130 K R K 157

Answers may vary. Sample: Elevation near a coastline varies between 2 m below and 8 m above sea level.

2.5 R x R 7.5 7 R x R 49 11 R x R 216 R x R 30

95908580 100 105 110 115

tepid warm hot



GPS



56. An emergency vehicle responding to a 911 call for a heart attack victimtraveled 5 miles to the patient’s home and then delivered him to thehospital 10 miles away. Which graph below represents the possible distancesthe emergency vehicle was from the hospital when the call was received? B

A.

B.

C.

D.

57. Which value below is a solution of neither 23x 2 7 $ 8 nor -2x 2 11 # 231?F. –6 G. 0 H. 10 J. 16

58. The County Water Department charges a monthly administration fee of$10.40 plus $.0059 for each gallon g of water used, up to 7,500 gallons.Find the minimum and maximum water consumption (in gallons) forcustomers whose monthly charge is at least $35 but no more than $50.Express amounts to the nearest gallon. Show your work. See left.

Mixed Review


59. 5 , 6b + 3 b S 60. 12n # 3n + 27 n K 3 61. 2 + 4r $ 5(r - 1)

Solve. If the equation is an identity or if it has no solution, write identity or no solution.

62. x - 3 = 5x + 1 –1 63. 4(w + 3) = 10w 2 64. 8p - 4 = 4(2p - 1)

Solve each inequality. Graph the solution. 1–6. See margin.

1. 8d + 2 , 5d - 7 2. 2n + 1 $ -3 3. -1 # 4m + 7 # 11

4. 5s - 3 + 1 , 8 5. 5(3p - 2) . 50 6. 3 - x $ 7 or 2x - 3 . 5

Write an inequality that represents each situation.

7. A cat weighs less than 8 pounds. c R 8

8. We expect today’s temperature to be between 658F and 758F, inclusive.

9. Geometry The length of each side of a rectangular picture frame needs to be15 in. You have only one 48 in. piece of wood to use for this frame. Write andsolve an inequality that describes the possible widths for this frame.

10. Solve -2x + 7 # 45. x L –19

Lesson 3-3

13

Lesson 4-4

Short Response

6 8 10 12 14 16 18420

6 8 10 12 14 16 18420

6 8 10 12 14 16 18420

6 8 10 12 14 16 18420

Multiple Choice

Test Prep



58. [2] 35 K 10.4 ± 0.0059g K 50

24.6 K 0.0059g K 39.64169 R g R 6712

minimum consumption: 4169 galmaximum consumption: 6712 gal(OR equivalent explanation)

[1] incorrect answer ORinsufficient explanation

r K 7

identity

65 K t K 75

2(15) ± 2(w) K 48, w K 9

G

GO forHelp

232

Test Prep


Transparencies

Exercise 57 Suggest to studentsthat they first solve eachinequality.


26.

27.

28.

page 232 Checkpoint Quiz 2

1. d R –3

2. n L –2

3. –2 K m K 1

4. s R 2

�1 10 32

�3 �1�2 10 2

�3 �1�2 10

�4�3�2�5 1�1 0

�2 0�1 21 3

�3 �2�1�5�4 20 1

�4�2 0 2 4 86

5. p S 4

6. x K –4 or x S 4

�4�2 0�6 62 4

0 2 4�2 6 8

Use this Checkpoint Quiz to checkstudents’ understanding of theskills and concepts of Lessons 4-4through 4-5.

ResourcesGrab & Go• Checkpoint Quiz 2

Lesson 4-6 Absolute Value Equations and Inequalities 235

Absolute Value Equations andInequalities

Part 1

Recall that the absolute value of a number is its distance from zero on a numberline. Since absolute value represents distance, it can never be negative.

The graph of ΔxΔ = 3 is below.

The two solutions of the equation ΔxΔ = 3 are -3 and 3.

You can use the properties of equality to solve an absolute value equation.

Solving an Absolute Value Equation

Solve ΔxΔ + 5 = 11.

ΔxΔ + 5 - 5 = 11 - 5 Subtract 5 from each side.

ΔxΔ = 6 Simplify.

x = 6 or x = -6 Definition of absolute value.

Check ΔxΔ + 5 = 11

Δ6Δ + 5 0 11 d Substitute 6 and –6 for x. S Δ-6Δ + 5 0 11

6 + 5 = 11 ✓ 6 + 5 = 11 ✓

Solve each equation. Check your solution.a. ΔtΔ - 2 = -1 –1, 1 b. 3ΔnΔ = 15 –5, 5 c. 4 = 3ΔwΔ - 2 –2, 2d. Critical Thinking Is there a solution of 2ΔnΔ = -15? Explain.

No; an absolute value cannot be negative.

11Quick Check

EXAMPLEEXAMPLE11

�4 �3 �2 �1 0 1 2 3

3 units 3 units

4�5 5

4-64-6

Lessons 1-3 and 2-1

Simplify.

1. Δ15Δ 15 2. Δ-3Δ 3 3. Δ18 - 12Δ 6

4.-Δ-7Δ –7 5. Δ12 - (-12)Δ 24 6. Δ-10 + 8Δ 2

Complete each statement with ,, =, or ..

7. Δ3 - 7Δ ■ 4 8. Δ-5Δ + 2 ■ 6 9. Δ7Δ - 1 ■ 8

10. 6 - 2 ■ 3 11. -4 + 2 ■ 2 12. -3 - 4 ■ 7 58u1218u1

213u23u5

8u14u

What You’ll Learn• To solve equations that

involve absolute value

• To solve inequalities thatinvolve absolute value

. . . And WhyTo find a range of acceptablemeasurements for parts of anengine, as in Example 4

11 Solving Absolute Value Equations

≠

S

S R

S ≠

Find the numbers thatare 3 units from 0.


Problem Solving Hint

u 23 u 5 3

u 3 u 5 3

4-64-6

235

1. PlanObjectives1 To solve equations that

involve absolute value2 To solve inequalities that

involve absolute value

Examples1 Solving an Absolute Value

Equation2 Solving an Absolute Value

Equation3 Solving an Absolute Value

Inequality4 Real-World Problem Solving

Math Background

The rules for absolute valueequations and inequalities allproceed from the definition ofabsolute value.

Δx« = bMore Math Background: p. 198D





Exploring Real Numbers Lesson 1-3: Example 5Extra Skills and Word

Problem Practice, Ch. 1

Adding Rational NumbersLesson 2-1: Example 2Extra Skills and Word

Problem Practice, Ch. 2

x x $ 02x x , 0

PowerPoint

Special NeedsStudents may think that if , then .Reinforce the concept that absolute value refers todistance and cannot be negative.

Below LevelAsk students to explain in their own words and also give an example of when to solve an absolute value inequality using or or and in thecompound inequality.

L2u 8 u 5 28u 28 u 5 8

L1



Some absolute value equations such as Δ2p + 5Δ = 11 have variable expressionswithin the absolute value symbols. The expression inside the absolute valuesymbols can be either positive or negative.

Solving an Absolute Value Equation

Solve Δ2p + 5Δ = 11.

2p + 5 = 11 d Write two equations. S 2p + 5 = -11

2p + 5 - 5 = 11 - 5 d Subtract 5 from each side. S 2p + 5 - 5 = -11 - 5

2p = 6 2p = -16

= d Divide each side by 2. S =

p = 3 p = – 8

The value of p is 3 or -8.

Solve each equation. Check your solution.a. Δc - 2Δ = 6 –4, 8 b. –5.5 = Δt + 2Δ c. Δ7dΔ = 14 –2, 2

Part 2

You can write absolute value inequalities as compound inequalities.

The graphs below show two absolute value inequalities.

Δn - 1Δ , 3 Δn - 1Δ . 3

Δn - 1Δ , 3 represents all numbers Δn - 1Δ . 3 represents all numberswhose distance from 1 is less than whose distance from 1 is greater than 3 units. So -3 , n - 1 , 3. 3 units. So n - 1 , -3 or n - 1 . 3.

�3 �2 �1 0 1 2 3

3 units 3 units

4 5�3 �2 �1 0 1 2 3

3 units 3 units

4 5

22Quick Check

2162

2p2

62

2p2

EXAMPLEEXAMPLE22

12 Solving Absolute Value Inequalities

Key Concepts Rule Solving Absolute Value Equations

To solve an equation in the form ΔAΔ = b, where A represents a variableexpression and b . 0, solve A = b and A = -b.

Key Concepts Rule Solving Absolute Value Inequalities

To solve an inequality in the form ΔAΔ , b, where A is a variable expressionand b . 0, solve -b , A , b.

To solve an inequality in the form ΔAΔ . b, where A is a variable expressionand b . 0, solve A , -b or A . b.

Similar rules are true for ΔAΔ # b or ΔAΔ $ b.

no solution

236

2. Teach

Guided Instruction

Math Tip

Stress to students that theabsolute value term must bealone on one side of the equalsign before writing an absolutevalue equation as two separateequations.

Technology Tip

Students can use a graphingcalculator to check their solutions.

Additional Examples

Solve Δa« - 3 = 5. a ≠ 8 or a ≠ –8

Solve Δ3c - 6« = 9. c ≠ 5 or c ≠ –1.

Tactile Learners

Some students may think thatonly -1 and 7 are solutions. Havestudents graph the solution ontheir own paper. Instruct studentsto place a finger on -1, and thenrun the finger along the numberline to 7. Then have them writeall integers that their fingertouched. Stress that all of theintegers they wrote and all thereal numbers in between themare solutions of the inequality.

EXAMPLEEXAMPLE33

22

11

EXAMPLEEXAMPLE22

EXAMPLEEXAMPLE11

Advanced LearnersAsk students to write a real-world problem that canbe modeled using an absolute value inequality. Havestudents exchange problems and discuss theiranswers.

English Language Learners ELL Some students may not be familiar with the wordpiston. Have a student who knows about internalcombustion engines explain what a piston is, how itworks, and perhaps draw one for other students to see.

L4


PowerPoint


Solving an Absolute Value Inequality

Solve Δv - 3Δ $ 4. Graph the solutions.

v - 3 # -4 or v - 3 $ 4 Write a compound inequality.

v - 3 + 3 # -4 + 3 P v - 3 + 3 $ 4 + 3 Add 3.

v # -1 or v $ 7 Simplify.

a. Solve and graph Δw + 2Δ . 5. See below.b. Critical Thinking What are the solutions of Δw + 2Δ . -5? all real numbers

To maintain quality, a manufacturer sets limits for how much an item can vary fromits specifications. You can use an absolute value equation to model a quality-control situation.

Manufacturing The ideal diameter of a piston for one type of car engine is90.000 mm. The actual diameter can vary from the ideal by at most 0.008 mm.Find the range of acceptable diameters for the piston.

Relate

Define Let d = actual diameter in millimeters of the cylindrical part.

Write

Δd - 90.000Δ # 0.008

-0.008 # d - 90.000 # 0.008 Write a compound inequality.

-0.008 + 90.000 # d - 90.000 + 90.000 # 0.008 + 90.000 Add 90.000.

89.992 # d # 90.008 Simplify.

The actual diameter must be between 89.992 mm and 90.008 mm, inclusive.

The ideal weight of one type of model airplane engine is 33.86 ounces. The actualweight may vary from the ideal by at most 0.05 ounce. Find the range of acceptableweights for this engine. 33.81 oz to 33.91 oz, inclusive


Solve each equation. If there is no solution, write no solution.

1. ΔbΔ = 2 –2, 2 2. 4 = ΔyΔ –4, 4 3. ΔwΔ = – ,

4. ΔnΔ + 2 = 8 –6, 6 5. 7 = ΔsΔ + 4 –3, 3 6. ΔxΔ - 10 = -3 –7, 7

7. 4ΔdΔ = 20 –5, 5 8. -3ΔmΔ = -6 –2, 2 9. ΔyΔ + 3 = 3 0

10. 12 = -4ΔkΔ 11. 2ΔzΔ - 5 = 1 –3, 3 12. 16 = 5ΔpΔ - 4 –4, 4

12

12

12Example 1

(page 235)

44Quick Check

0.008 mm#Δd - 90.000Δ

0.008 mmis at mostdifference between actualand ideal


33Quick Check0 1 2 3 4 5 6 7 8�2 �1

EXAMPLEEXAMPLE33



ConnectionReal-World ConnectionReal-World

Careers A quality-controlinspector inspects products tomaintain quality. For enginesproduced on an assembly line,an inspector selects engines atrandom to check quality ofmaterials and manufacturing.

�4 �2 0�8�6 62 4a. w R –7 or w S 3,

no solution

GO forHelp

237

Additional Examples

Solve Δy - 5« # 2. Graph thesolutions. 3 K y K 7

The ideal diameter of a pistonfor one type of car is 88.000 mm.The actual diameter can vary fromthe ideal by at most 0.007 mm.Find the range of acceptablediameters for the piston. between87.993 mm and 88.007 mm,inclusive


Adapted Instruction

Closure

Ask students to explain when touse and and when to use or in anabsolute value inequality.Assuming that the absolute valueexpression is on the left side ofthe inequality, if the inequalityreads is less than or is less thanor equal to, use and. If it reads isgreater than or is greater than orequal to, use or.

L1

L3

44

�1�1 00 11 22 33 44 10109988776655

33

PowerPoint

Solve each equation. If there is no solution, write no solution.

13. Δr - 8Δ = 5 3, 13 14. Δc + 2Δ = 6 –8, 4 15. 2 = Δg + 1Δ –3, 1

16. 3 = Δm + 2Δ –5, 1 17. Δv - 2Δ = 7 –5, 9 18. -3Δy - 3Δ = 9

19. 2Δd + 3Δ = 8 –7, 1 20. -2Δ7dΔ = -14 –1, 1 21. 1.2Δ5pΔ = 3.6

22. Complete each statement with less than or greater than.a. For ΔxΔ , 5, the graph includes all points whose distance is 9 5 units

from 0. less thanb. For ΔxΔ . 5, the graph includes all points whose distance is 9 5 units

from 0. greater than

Solve each inequality. Graph your solution. 23–34. See margin.

23. ΔkΔ . 2.5 24. ΔwΔ , 2 25. Δx + 3Δ , 5

26. Δn + 8Δ $ 3 27. Δy - 2Δ # 1 28. Δp - 4Δ # 3

29. Δ2c - 5Δ , 9 30. Δ2y - 3Δ $ 7 31. Δ3t + 1Δ . 8

32. Δ4x + 1Δ . 11 33. Δ5t - 4Δ $ 16 34. Δ3 - rΔ , 5

35. Manufacturing The ideal diameter of a gear for a certain type of clock is12.24 mm. An actual diameter can vary by 0.06 mm. Find the range ofacceptable diameters. between 12.18 mm and 12.30 mm, inclusive

36. Manufacturing The ideal width of a certain conveyor belt for a manufacturingplant is 50 in. An actual conveyor belt can vary from the ideal by at most in.Find the acceptable widths for this conveyor belt.

Solve each equation or inequality.

37. Δ2dΔ + 3 = 21 –9, 9 38. Δ–3nΔ - 2 = 7 –3, 3 39. ΔpΔ - = –1 , 1

40. ΔtΔ + 2.7 = 4.5 41. 4Δk + 1Δ = 16 –5, 3 42. -2Δc - 4Δ= -8 0, 8

43. Δ3dΔ $ 6 44. ΔnΔ - 3 . 7 45. 9 , Δc + 7Δ

46. = -4.2 –12.6, 12.6 47. Δ6.5xΔ , 39 48. 4ΔnΔ = 32 –8, 8

49. + 1 = 5 –8, 8 50. ΔaΔ + = –3, 3 51. 4 - 3Δm + 2Δ . -14

Write an absolute value inequality that represents each situation.

52. all numbers less than 3 units from 0 n R 3

53. all numbers more than 7.5 units from 0 n S 7.5

54. all numbers more than 2 units from 6 n – 6 S 2

55. all numbers at least 3 units from –1 n ± 1 L 3

56. Manufacturing A pasta manufacturer makes 16-ounce boxes of macaroni.The manufacturer knows that not every box weighs exactly 16 ounces. Theallowable difference is 0.05 ounce. Write and solve an absolute value inequalitythat represents this situation. w – 16 K 0.05, 15.95 K w K 16.05

57. Elections In a poll for the upcoming mayoral election, 42% of likely voterssaid they planned to vote for Lucy Jones. This poll has a margin of error of 4 3 percentage points. Use the inequality Δv - 42Δ # 3 to find the least andgreatest percent of voters v likely to vote for Lucy Jones according to this poll.

»»

»»

»»

»»

»»

3 12

12u1

2 au

| v |23

12

12

56

23

Apply Your SkillsBB

732

Example 4(page 237)

Example 3(page 237)

Example 2(page 236)

between 49 in. and

50 in., inclusive732

2532

no solution

–0.6, 0.6

c R –16 or c S 2d K –2 or d L 2 n R –10 or n S 10

–1.8, 1.8

–8 R m R 4


39%, 45%

–6 R x R 6

GPS

238


23. k R –2.5 or k S 2.5;

24. –2 R w R 2;

25. –8 R x R 2;

�2�4�6�8 0 2

�2 �1 0�3 31 2

�3�4 �2�1 0 1 2 3 4

26. n K –11 or n L –5;

27. 1 K y K 3;

0 1 2�1 53 4

�8 �6�4�12�10 2�2 0

28. 1 K p K 7;

29. –2 R c R 7;

�2 0 2�4 84 6

1 2 3�1 0 6 7 84 5

30. y K –2 or y L 5;

31. t R –3 or t S 2 ;

�3�4 �2�1 0 1 2 3 4

13

�4 0�2 42 6


A B 1-21, 37-42, 61-71

A B 22-36, 45-60C Challenge 72-80



Error Prevention!

Exercises 37–51 Remind studentsto isolate the absolute valuebefore writing the twoinequalities.

2

1


Enrichment

Reteaching

Adapted Practice

Name Class Date

© P

ears

on E

duc

atio

n, In

c. A

ll rig

hts

rese

rved

.

Practice 4-6 Probability of Compound Events

1. Suppose you have a dark closet containing seven blue shirts, five yellowshirts, and eight white shirts. You pick two shirts from the closet. Find each probability.

a. P(blue then yellow) with replacing b. P(blue then yellow) without replacing

c. P(yellow then yellow) with replacing d. P(yellow then yellow) without replacing

e. P(yellow then white) with replacing f. P(yellow then white) without replacing

g. P(blue then blue) with replacing h. P(blue then blue) without replacing

A and B are independent events. Find the missing probability.

2. P(A) = , P(A and B) = . Find P(B).

3. P(B) = , P(A and B) = . Find P(A).

4. P(B) = , P(A and B) = . Find P(A).

5. P(A) = , P(B) = . Find P(A and B).

6. Suppose you draw two tennis balls from a bag containing seven pink,four white, three yellow, and two striped balls. Find each probability.

a. P(yellow then pink) with replacing b. P(yellow then pink) without replacing

c. P(pink then pink) with replacing d. P(pink then pink) without replacing

e. P(striped then striped) with replacing f. P(striped then striped) without replacing

g. P(pink then white) with replacing h. P(pink then white) without replacing

A and B are independent events. Find the missing probability.

7. P(A) = , P(A and B) = . Find P(B).

8. P(A) = , P(B) = . Find P(A and B).

9. P(B) = , P(A and B) = . Find P(A).

10. P(B) = , P(A and B) = . Find P(A).

Use an equation to solve each problem.

11. A bag contains green and yellow color tiles. You pick two tiles withoutreplacing the first one. The probability that the first tile is yellow is .The probability of drawing two yellow tiles is . Find theprobability that the second tile you pick is yellow.

12. A bag contains red and blue marbles. You pick two marbles withoutreplacing the first one. The probability of drawing a blue and then a redis . The probability that your second marble is red if your first marbleis blue is . Find the probability that the first marble is blue.2

3

415

1235

35

320

14

35

910

16

37

12

34

34

815

34

1516

213

15

13

37

Practice

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L4

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239

32. x R –3 or x S 2.5;

33. t K –2.4 or t L 4;

�3�4 �2�1 0 1 2 3 4 5

�2 �1 0�4�3 31 2

34. –2 R r R 8;

0 2 4�4�2 106 8

70a. between 7.075 oz and7.105 oz, inclusive

b. No; the excess weight of some coins may bebalanced by the lowerweight of other coins.

Lesson Quiz

Solve.

1. Δa« + 6 = 9 a ≠ 3 ora ≠ –3

2. Δ2x + 3« = 7 x ≠ 2 orx ≠ –5

3. Δp + 6« # 1 –7 K p K –5

4. 3Δx + 4« � 15 x S 1 or x R –9


Have students make a list of thesteps to follow in order to solveany absolute value inequality.1. Isolate the absolute value.2. If S, write as follows:

expression R opposite numberor expression S originalnumber

3. If R, write as follows:opposite number R expressionR original number.

4. Solve the compound inequality.

PowerPoint

4. Assess & Reteach


58. Quality Control A box of one brand of crackers should weigh 454 g. Thequality-control inspector randomly selects boxes to weigh. The inspector sends back any box that is not within 5 g of the ideal weight.a. Write an absolute value inequality for this situation. w – 454 K 5b. What is the range of allowable weights for a box of crackers?

59. Gears Acceptable diameters for one type of gear are from 6.25 mm to6.29 mm. Write an absolute value inequality for the acceptable diametersfor the gear. g – 6.27 K 0.02

60. Writing Explain why the absolute value inequality Δ2c - 5Δ + 9 , 4 has no solution. The absolute value of a number cannot be less than zero.

61. Open-Ended Write an absolute value equation using the numbers 5, 3,-12.Then solve your equation. sample: 5x – 12 ≠ 3; 1 , 3

Write an absolute value equation that has the given values as solutions.

Sample 8, 2

Δx - 5Δ = 3 Since 8 and 2 are both 3 units from 5, write »x– 5» ≠ 3.

62. 2, 6 x – 4 ≠ 2 63. -2, 6 64. -3, 9 65. 9, 16

66. –1, 7 x – 3 ≠ 4 67. 3, 8 68. –15, –3 69. 2, 10

70. Banking The ideal weight of a nickel is 0.176 ounce. To check that there are 40 nickels in a roll, a bank weighs the roll and allows for an error of 0.015ounce in the total weight. See margin. a. What is the range of acceptable weights if the wrapper weighs 0.05 ounce?b. Critical Thinking For any given roll of nickels, can you be certain that all the

coins are acceptable? Explain.

71. a. Meteorology A meteorologist reported that the previous day’s temperaturesvaried 14 degrees from the normal temperature of 258F. What were themaximum and minimum temperatures possible on the previous day?

b. Write an absolute value equation for the temperature. t – 25 ≠ 14

Solve each equation. Check your solution.

72. Δx + 4Δ = 3x 2 73. Δ4x - 5Δ = 2x + 1 , 3 74. Δ2x + 3Δ = 4x 3

Replace the ■ with K,L, or ≠.

75. Δa + bΔ ■ ΔaΔ + ΔbΔ 76. Δa - bΔ ■ ΔaΔ - ΔbΔ

77. ΔabΔ ■ ΔaΔ?ΔbΔ 78. ■ , b 2 0

Write an absolute value inequality that each graph could represent.

79. 80.


81. Which compound inequality has the same meaning as Δx + 4Δ , 8? AA.-12 , x , 4 B. -12 . x . 4C. x , -12 or x . 4 D. x . -12 or x , 4

82. Which of the following values is a solution of Δ2 - xΔ , 4? GF. -2 G. -1 H. 6 J. 7

Multiple Choice

�4 0 2�6 4 6�2�4 0 2�6 4 6�2

Z ˛a˛ ZZ ˛b˛ Z P ab ˛

P

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ChallengeCC

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Test Prep

between 449 g and 459 g, inclusive

K L

≠ ≠

x – 2 ≠ 4»»

x – 5 ≠ 212»

12» x ± 9 ≠ 6»»

x – 3 ≠ 6»»

x – 6 ≠ 4»»

x – 12 ≠ 312»

12»

a. 11�F, 39�F

x ± 1 S 3»» x – 2 K 4»»




83. The ideal diameter of a metal rod for a lamp is 1.25 inches with an allowableerror of at most 0.005 inch. Which rod below would not be suitable? DA. a rod with diameter 1.249 inchesB. a rod with diameter 1.251 inchesC. a rod with diameter 1.253 inchesD. a rod with diameter 1.355 inches

84. A delivery driver receives a bonus if he delivers pizza to a customer in30 minutes plus or minus 5 minutes. Which inequality or equationrepresents the driver’s allotted time to receive a bonus? J

F. Δx - 30Δ , 5 G. Δx - 30Δ . 5

H. Δx - 30Δ = 5 J. Δx - 30Δ # 5

85. Water is in a liquid state if its temperature t, in degrees Fahrenheit,satisfies the inequality Δt - 122Δ , 90. Which graph represents thetemperatures described by this inequality? A

A. B.

C. D.

86. A bicycling club is planning a trip. The graphs below show the number ofmiles three people want to cycle per day.

Ramon

Kathleen

Allan

a. Draw a graph showing a trip length that would be acceptable to allthree bikers. See above left.

b. Explain how your graph relates to the graphs above. See above left.

Mixed Review

Write a compound inequality to model each situation.

87. Elevation in North America is between the highest elevation of 20,320 ft abovesea level at Mount McKinley, Alaska, and the lowest elevation of 282 ft belowsea level at Death Valley, California. –282 K e K 20,320

88. Normal body temperature t is within 0.6 degrees of 36.6°C.


89. 3t + 4t = -21 –3 90. 9(-2n + 3) = -27 3 91. k + 5 - 4k = -10 5

92. 5x + 3 - 2x = -21 –8 93. 5.4m - 2.3 = -0.5 94. 3(y - 4) = 9 7

Write each group of numbers from least to greatest.

95. 3,-2, 0,-2.5,p –2.5, –2, 0, 3, π 96. ,-1.5, , 7,-2 –2, –1.5, – , 7,

97. 0.001, 0.01, 0.009, 0.011 98. –p, 2p, –2.5, –3, 3

152

4324

3152

Lesson 1-3

13

Lesson 3-2

Lesson 4-5

20 25 30 35 4015105

20 25 30 35 4015105

20 25 30 35 4015105

Short Response

90 12290 122

32 21232 212


86a. [2]

86b. [2] The overlap of thethree graphs is from20 to 25, inclusive.

86a–b. [1] no graph ORinsufficientexplanation

15 2520 3530

Let t ≠ body temperature (�C), 36.0 K t K 37.2.

0.001, 0.009, 0.01, 0.011 –π, –3, –2.5, 3, 2π

GO forHelp

240

Test Prep


Transparencies

Error Prevention!

Exercise 82 Suggest studentsrewrite 2 - x as -x + 2 beforesolving.

4-1 inequalities and their graphs - weebly

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