4-5 Compound Inequalities

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

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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

Check Skills You’ll Need GO for Help

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

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:

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

Problems Practice, Ch. 1

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

Problems Practice, Ch. 4

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

228 Chapter 4 Solving Inequalities

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

EXAMPLEEXAMPLE Real-World Problem Solving33

22Quick Check

0 1 2 3 4 5�5 �4 �3 �2 �1

EXAMPLEEXAMPLE22

ConnectionReal-World

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;

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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

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22

23 24 25 26 27 28 3332313029

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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

learning style: verbal learning style: verbal

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..

Practice and Problem Solving

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)

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2622

22v22

284

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EXAMPLEEXAMPLE55

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EXAMPLEEXAMPLE44

12 Solving Compound Inequalities Joined by Or

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

Practice by ExampleAA

–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

Resources• Daily Notetaking Guide 4-5• Daily Notetaking Guide 4-5—

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

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0 10 10090807050 60403020

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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

5–19. See margin for graphs.

24–33. See margin for graphs.

�2 0 2�4 4 6

0 2 4�2 6 8

pages 229–232 Exercises

5.

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�10 �5 0 5 10

7.

8.

9.�1 10 32 4

�4 �3�2�5 1�1 0

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11.

12.�3�2�1 0 1 2

4 65 87 9

�1�2 0 1 2 3 4

3. PracticeAssignment Guide

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

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

Test Prep 56-58Mixed Review 59-64

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

Guided Problem SolvingGPS

Enrichment

Reteaching

Adapted Practice

Name Class Date

© P

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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.

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2 4 6�2 0 128 1023.

24.

25.0 21 43 5

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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

Alternative Assessment

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.

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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

ConnectionReal-World

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

lesson quiz, PHSchool.com, Web Code: ata-0405

GO nlineHomework HelpVisit: PHSchool.comWeb Code: ate-0405

GPS


Standardized Test Prep

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

Solve each inequality.

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

Mixed ReviewMixed Review

Checkpoint Quiz 2 Lessons 4-4 through 4-5

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

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

Transparencies

Exercise 57 Suggest to studentsthat they first solve eachinequality.

pages 229–232 Exercises

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

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�3 �1�2 10

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5. p S 4

6. x K –4 or x S 4

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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

Documents

4-5 Compound Inequalities