Absolute Value Equations and Inequalities

4-6

4-6 1. Plan

What You’ll Learn

Check Skills You’ll Need

• To solve equations that

Simplify.

involve absolute value

• To solve inequalities that involve absolute value

GO for Help

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

To find a range of acceptable measurements for parts of an engine, as in Example 4

10. 6 - 2 14 ■ 3 58 S

u

11. -4 23 + 2 13 ■ 2 12 S

u

u

u

Objectives 1 2

R 9. Δ7Δ - 1 ■ 8

1

12. -3 18 - 4 12 ■ 7 58 ≠

u

u

2 3 4

1

1 1 Solving Absolute Value Equations Part The graph of ΔxΔ = 3 is below. 3 units

u 3u 5 3 u 23u 5 3

Find the numbers that are 3 units from 0.

Δx« = b

Solving an Absolute Value Equation

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

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

ΔxΔ = 6 x=6

or

PowerPoint

Simplify.

x = -6

Bell Ringer Practice

Definition of absolute value.

Check Skills You’ll Need

Check ΔxΔ + 5 = 11 Δ6Δ + 5 0 11

d Substitute 6 and –6 for x. S

6 + 5 = 11 ✓

Quick Check

For intervention, direct students to:

Δ-6Δ + 5 0 11

Exploring Real Numbers

6 + 5 = 11 ✓

1 Solve each equation. Check your solution. a. ΔtΔ - 2 = -1 –1, 1 b. 3ΔnΔ = 15 –5, 5 c. 4 = 3ΔwΔ - 2 –2, 2 d. Critical Thinking Is there a solution of 2ΔnΔ = -15? Explain. No; an absolute value cannot be negative. Lesson 4-6 Absolute Value Equations and Inequalities

Special Needs

x$0 x,0

Lesson Planning and Resources

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

ΔxΔ + 5 - 5 = 11 - 5

x 2x

More Math Background: p. 198D

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

EXAMPLE

The rules for absolute value equations and inequalities all proceed from the definition of absolute value.

3 units

5 4 3 2 1 0 1 2 3 4 5

1

Solving an Absolute Value Equation Solving an Absolute Value Equation Solving an Absolute Value Inequality Real-World Problem Solving

Math Background

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

To solve equations that involve absolute value To solve inequalities that involve absolute value

Examples

Complete each statement with ,, =, or .. ≠ S 7. Δ3 - 7Δ ■ 4 8. Δ-5Δ + 2 ■ 6

. . . And Why

Lessons 1-3 and 2-1

Below Level

L1

Students may think that if u 28 u 5 8, then u 8 u 5 28. Reinforce the concept that absolute value refers to distance and cannot be negative.

learning style: verbal

235

Lesson 1-3: Example 5 Extra Skills and Word Problem Practice, Ch. 1

Adding Rational Numbers Lesson 2-1: Example 2 Extra Skills and Word Problem Practice, Ch. 2

L2

Ask 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 the compound inequality.

learning style: verbal

235

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

2. Teach Guided Instruction Key Concepts 1

EXAMPLE

Rule

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

Math Tip

Stress to students that the absolute value term must be alone on one side of the equal sign before writing an absolute value equation as two separate equations.

2

EXAMPLE

Solving Absolute Value Equations

2

EXAMPLE

Solving an Absolute Value Equation

Solve Δ2p + 5Δ = 11.

Technology Tip

Students can use a graphing calculator to check their solutions.

2 p + 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

2p 6 2 =2

PowerPoint

2p 216 2 = 2

d Divide each side by 2. S

p=3

Additional Examples

p = –8

The value of p is 3 or -8.

1 Solve Δa« - 3 = 5. a ≠ 8

Quick Check

or a ≠ –8 2 Solve Δ3c - 6« = 9. c ≠ 5

2 Solve each equation. Check your solution. a. Δc - 2Δ = 6 –4, 8 b. –5.5 = Δt + 2Δ no solution

c. Δ7dΔ = 14 –2, 2

or c ≠ –1.

3

EXAMPLE

Tactile Learners

Some students may think that only -1 and 7 are solutions. Have students graph the solution on their own paper. Instruct students to place a finger on -1, and then run the finger along the number line to 7. Then have them write all integers that their finger touched. Stress that all of the integers they wrote and all the real numbers in between them are solutions of the inequality.

2

1 2 Solving Absolute Value Inequalities Part You can write absolute value inequalities as compound inequalities. The graphs below show two absolute value inequalities. Δn - 1Δ , 3 3 units

Δn - 1Δ . 3

3 units

3 units

3 2 1 0 1 2 3 4 5

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

Key Concepts

Rule

3 units

3 2 1 0 1 2 3 4 5

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

Solving Absolute Value Inequalities

To solve an inequality in the form ΔAΔ , b, where A is a variable expression and b . 0, solve -b , A , b. To solve an inequality in the form ΔAΔ . b, where A is a variable expression and b . 0, solve A , -b or A . b. Similar rules are true for ΔAΔ # b or ΔAΔ $ b.

236

Chapter 4 Solving Inequalities

Advanced Learners

English Language Learners ELL

L4

Ask students to write a real-world problem that can be modeled using an absolute value inequality. Have students exchange problems and discuss their answers.

236

learning style: verbal

Some students may not be familiar with the word piston. Have a student who knows about internal combustion engines explain what a piston is, how it works, and perhaps draw one for other students to see. learning style: verbal

PowerPoint

3

Solving an Absolute Value Inequality

EXAMPLE

Additional Examples

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

or

v-3$4

v - 3 + 3 # -4 + 3

P

v-3+3$4+3

v # -1

v$7

or

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

Write a compound inequality. Add 3.

1 0 1 2 3 4 5 6 7 8 9 10

Simplify.

2 1 0 1 2 3 4 5 6 7 8

Quick Check

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

a. w R –7 or w S 3,

8 6 4 2 0 2 4 6 To maintain quality, a manufacturer sets limits for how much an item can vary from its specifications. You can use an absolute value equation to model a qualitycontrol situation.

4

Real-World

EXAMPLE

Problem Solving

Manufacturing The ideal diameter of a piston for one type of car engine is 90.000 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

difference between actual and ideal

is at most

Write Connection

Careers A quality-control inspector inspects products to maintain quality. For engines produced on an assembly line, an inspector selects engines at random to check quality of materials and manufacturing.

Quick Check

Δd - 90.000Δ

#

0.008 mm

0.008 mm

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 89.992 #

d

# 90.008

Resources • Daily Notetaking Guide 4-6 L3 • Daily Notetaking Guide 4-6— L1 Adapted Instruction

Closure

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

4 The ideal diameter of a piston for one type of car is 88.000 mm. The actual diameter can vary from the ideal by at most 0.007 mm. Find the range of acceptable diameters for the piston. between 87.993 mm and 88.007 mm, inclusive

Add 90.000. Simplify.

Ask students to explain when to use and and when to use or in an absolute value inequality. Assuming that the absolute value expression is on the left side of the inequality, if the inequality reads is less than or is less than or equal to, use and. If it reads is greater than or is greater than or equal to, use or.

The actual diameter must be between 89.992 mm and 90.008 mm, inclusive. 4 The ideal weight of one type of model airplane engine is 33.86 ounces. The actual weight may vary from the ideal by at most 0.05 ounce. Find the range of acceptable weights for this engine. 33.81 oz to 33.91 oz, inclusive

EXERCISES

For more exercises, see Extra Skill and Word Problem Practice.

Practiceand andProblem ProblemSolving Solving Practice A

Practice by Example Example 1

GO for Help

(page 235)

Solve each equation. If there is no solution, write no solution. 1. ΔbΔ = 2 –2, 2

2. 4 = ΔyΔ –4, 4

4. ΔnΔ + 2 = 8 –6, 6

5. 7 = ΔsΔ + 4 –3, 3

3. ΔwΔ = 12 –12 , 12 6. ΔxΔ - 10 = -3 –7, 7

7. 4ΔdΔ = 20 –5, 5

8. -3ΔmΔ = -6 –2, 2

9. ΔyΔ + 3 = 3 0

10. 12 = -4ΔkΔ no solution

11. 2ΔzΔ - 5 = 1 –3, 3

12. 16 = 5ΔpΔ - 4 –4, 4

Lesson 4-6 Absolute Value Equations and Inequalities

237

237

3. Practice

Example 2 (page 236)

Assignment Guide 1 A B 1-21, 37-42, 61-71 Example 3

2 A B

22-36, 45-60 C Challenge 72-80 Test Prep Mixed Review

(page 237)

81-86 87-98

Solve each equation. If there is no solution, write no solution. 13. Δr - 8Δ = 5 3, 13

14. Δc + 2Δ = 6 –8, 4

16. 3 = Δm + 2Δ –5, 1

17. Δv - 2Δ = 7 –5, 9

15. 2 = Δg + 1Δ –3, 1

18. -3Δy - 3Δ = 9 no solution 19. 2Δd + 3Δ = 8 –7, 1 20. -2Δ7dΔ = -14 –1, 1 21. 1.2Δ5pΔ = 3.6 –0.6, 0.6 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 than b. 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.

Homework Quick Check To check students’ understanding of key skills and concepts, go over Exercises 13, 35, 52, 57, 60.

Error Prevention! Exercises 37–51 Remind students

Example 4

to isolate the absolute value before writing the two inequalities.

(page 237)

23. Δ kΔ . 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 is 12.24 mm. An actual diameter can vary by 0.06 mm. Find the range of acceptable diameters. between 12.18 mm and 12.30 mm, inclusive 36. Manufacturing The ideal width of a certain conveyor belt for a manufacturing 7 plant is 50 in. An actual conveyor belt can vary from the ideal by at most 32 in. 25 Find the acceptable widths for this conveyor belt. between 49 32 in. and

B

Apply Your Skills

37. Δ2dΔ + 3 = 21 –9, 9 GPS Guided Problem Solving

L4 L2

Reteaching

L1

Adapted Practice Practice Name

Class

Practice 4-6

u u

L3

Date

38. Δ–3nΔ - 2 = 7 –3, 3

40. ΔtΔ + 2.7 = 4.5 41. 4Δk + 1Δ = 16 –5, 3 –1.8, 1.8 43. Δ3dΔ $ 6 44. ΔnΔ - 3 . 7 d K –2 or d L 2 n R –10 or n S 10 v | = -4.2 –12.6, 12.6 47. Δ6.5xΔ , 39 46. |23 –6 R x R 6 49. 12 a + 1 = 5 –8, 8 50. ΔaΔ + 21 = 3 12 –3, 3

L3

Enrichment

7 50 32 in., inclusive

Solve each equation or inequality.

Probability of Compound Events

1. Suppose you have a dark closet containing seven blue shirts, five yellow shirts, 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

39. ΔpΔ - 32 = 56 –1 21 , 1 12 42. -2Δc - 4Δ= -8 0, 8 45. 9 , Δc + 7Δ c R –16 or c S 2 48. 4ΔnΔ = 32 –8, 8 51. 4 - 3Δm + 2Δ . -14 –8 R m R 4

Write an absolute value inequality that represents each situation. 52. all numbers less than 3 units from 0 »n» R 3

A and B are independent events. Find the missing probability.

2. P(A) = 37 , P(A and B) = 31 . Find P(B). 2 . Find P(A). 3. P(B) = 51 , P(A and B) = 13 15 , P(A and B) = 3 . Find P(A). 4. P(B) = 16 4 8 , P(B) = 3 . Find P(A and B). 5. P(A) = 15 4

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

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

55. all numbers at least 3 units from –1 »n ± 1» L 3

© Pearson Education, Inc. All rights reserved.

A and B are independent events. Find the missing probability.

7. P(A) = 34 , P(A and B) = 21 . Find P(B). 8. P(A) = 37 , P(B) = 16 . Find P(A and B). 9 , P(A and B) = 3 . Find P(A). 9. P(B) = 10 5 3 . Find P(A). 10. P(B) = 41 , P(A and B) = 20

56. Manufacturing A pasta manufacturer makes 16-ounce boxes of macaroni. The manufacturer knows that not every box weighs exactly 16 ounces. The allowable difference is 0.05 ounce. Write and solve an absolute value inequality that represents this situation. »w – 16» K 0.05, 15.95 K w K 16.05

Use an equation to solve each problem.

11. A bag contains green and yellow color tiles. You pick two tiles without replacing the first one. The probability that the first tile is yellow is 53. The probability of drawing two yellow tiles is 12 35 . Find the probability that the second tile you pick is yellow. 12. A bag contains red and blue marbles. You pick two marbles without replacing the first one. The probability of drawing a blue and then a red 4 is 15 . The probability that your second marble is red if your first marble is blue is 23. Find the probability that the first marble is blue.

pages 237–240

57. Elections In a poll for the upcoming mayoral election, 42% of likely voters GPS said 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 and greatest percent of voters v likely to vote for Lucy Jones according to this poll. 39%, 45%

Exercises

23. k R –2.5 or k S 2.5; 4 3 21 0 1 2 3 4

24. –2 R w R 2;

12108 6 4 2 0 2

27. 1 K y K 3;

25. –8 R x R 2;

238

Chapter 4 Solving Inequalities

26. n K –11 or n L –5;

3 2 1 0 1 2 3

8 6 4 2

238

0

2

1 0 1 2 3 4 5

28. 1 K p K 7; 1 0 1 2 3 4 5 6 7 8

29. –2 R c R 7; 4 2 0 2 4 6 8

30. y K –2 or y L 5; 4 2 0 2 4 6

31. t R –3 or t S 2 13 ; 4 3 21 0 1 2 3 4

GO

nline

Homework Help Visit: PHSchool.com Web Code: ate-0406

4. Assess & Reteach

58. Quality Control A box of one brand of crackers should weigh 454 g. The quality-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 5 b. What is the range of allowable weights for a box of crackers? between 449 g and 459 g, inclusive 59. Gears Acceptable diameters for one type of gear are from 6.25 mm to 6.29 mm. Write an absolute value inequality for the acceptable diameters for the gear. »g – 6.27» K 0.02

PowerPoint

Lesson Quiz Solve.

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 45 , 3 Write an absolute value equation that has the given values as solutions.

1. Δa« + 6 = 9 a ≠ 3 or a ≠ –3 2. Δ2x + 3« = 7 x ≠ 2 or x ≠ –5 3. Δp + 6« # 1 –7 K p K –5 4. 3Δx + 4«  15 x S 1 or x R –9

Sample 8, 2 Δx - 5Δ = 3 Since 8 and 2 are both 3 units from 5, write »x – 5» ≠ 3.

Alternative Assessment

62. 2, 6 »x – 4» ≠ 2 63. -2, 6 64. -3, 9 65. 9, 16 »x – 2» ≠ 4 »x – 3» ≠ 6 »x – 121 » ≠ 312 66. –1, 7 »x – 3» ≠ 4 67. 3, 8 68. –15, –3 69. 2, 10 2 »x – 512 » ≠ 212 »x ± 9» ≠ 6 »x – 6» ≠ 4 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 0.015 ounce 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.

C

Challenge

71. a. Meteorology A meteorologist reported that the previous day’s temperatures varied 14 degrees from the normal temperature of 258F. What were the maximum and minimum temperatures possible on the previous day? b. Write an absolute value equation for the temperature. »t – 25» ≠ 14 a. 11F, 39F Solve each equation. Check your solution. 72. Δx + 4Δ = 3x 2 73. Δ4x - 5Δ = 2x + 1 2 , 3 74. 4Δ2x + 3Δ = 4x 3 3

3

Replace the ■ with K, L, or ≠. K 75. Δa + bΔ ■ ΔaΔ + ΔbΔ

Have students make a list of the steps to follow in order to solve any absolute value inequality. 1. Isolate the absolute value. 2. If S, write as follows: expression R opposite number or expression S original number 3. If R, write as follows: opposite number R expression R original number. 4. Solve the compound inequality.

L 76. Δa - bΔ ■ ΔaΔ - ΔbΔ

Za Z 78. P ba P ■ Z b Z , b 2 0 ≠ Write an absolute value inequality that each graph could represent.

77. ΔabΔ ■ ΔaΔ?ΔbΔ ≠ 79.

6 4 2 »x ± 1» S 3

0

˛

˛

˛

˛

˛

2

4

6

80.

6 4 2 »x – 2» K 4

0

2

4

6

StandardizedTest TestPrep Prep Multiple Choice

81. Which compound inequality has the same meaning as Δx + 4Δ , 8? A A. -12 , x , 4 B. -12 . x . 4 C. x , -12 or x . 4 D. x . -12 or x , 4 82. Which of the following values is a solution of Δ2 - xΔ , 4? G F. -2 G. -1 H. 6 J. 7

lesson quiz, PHSchool.com, Web Code: ata-0406

32. x R –3 or x S 2.5; 4 3 2 1 0 1 2 3

33. t K –2.4 or t L 4; 4 3 21 0 1 2 3 4 5

Lesson 4-6 Absolute Value Equations and Inequalities

34. –2 R r R 8; 4 2 0 2 4 6 8 10

239

70a. between 7.075 oz and 7.105 oz, inclusive b. No; the excess weight of some coins may be balanced by the lower weight of other coins.

239

Test Prep

83. The ideal diameter of a metal rod for a lamp is 1.25 inches with an allowable error of at most 0.005 inch. Which rod below would not be suitable? D A. a rod with diameter 1.249 inches B. a rod with diameter 1.251 inches C. a rod with diameter 1.253 inches D. a rod with diameter 1.355 inches

Resources For additional practice with a variety of test item formats: • Standardized Test Prep, p. 247 • Test-Taking Strategies, p. 242 • Test-Taking Strategies with Transparencies

84. A delivery driver receives a bonus if he delivers pizza to a customer in 30 minutes plus or minus 5 minutes. Which inequality or equation represents the driver’s allotted time to receive a bonus? J

Error Prevention! Exercise 82 Suggest students rewrite 2 - x as -x + 2 before solving.

86a. [2]

15 20 25 30 35

86b. [2] The overlap of the three graphs is from 20 to 25, inclusive. 86a–b. [1] no graph OR insufficient explanation Short Response

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 the temperatures described by this inequality? A A.

32

C.

B.

212

32

212

D.

90 122

90 122

86. A bicycling club is planning a trip. The graphs below show the number of miles three people want to cycle per day. Ramon Kathleen Allan

5

10

15

20

25

30

35

40

5

10

15

20

25

30

35

40

5

10

15

20

25

30

35

40

a. Draw a graph showing a trip length that would be acceptable to all three bikers. See above left. b. Explain how your graph relates to the graphs above. See above left.

Mixed Review

GO for Help

Lesson 4-5

Write a compound inequality to model each situation. 87. Elevation in North America is between the highest elevation of 20,320 ft above sea level at Mount McKinley, Alaska, and the lowest elevation of 282 ft below sea level at Death Valley, California. –282 K e K 20,320

Lesson 3-2

88. Normal body temperature t is within 0.6 degrees of 36.6°C. Let t ≠ body temperature (C), 36.0 K t K 37.2. Solve each equation. 89. 3t + 4t = -21 –3

90. 9(-2n + 3) = -27 3

92. 5x + 3 - 2x = -21 –8 93. 5.4m - 2.3 = -0.5 1 3 Lesson 1-3

240

240

91. k + 5 - 4k = -10 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

15 4 4 96. 15 2 , -1.5, 23, 7, -2 –2, –1.5, –3 , 7, 2

97. 0.001, 0.01, 0.009, 0.011 0.001, 0.009, 0.01, 0.011

98. –p, 2p, –2.5, –3, 3 –π, –3, –2.5, 3, 2π

Chapter 4 Solving Inequalities