2 Solving Absolute Value Inequalities

Solving Absolute Value Inequalities

EXAMPLE Solving Absolute Value Inequalities11

a. Solve ∣ x + 6 ∣ ≤ 3. Graph the solution. 3. Graph the solution.

Use ∣ x + 6 ∣ ≤ 3 to write two related inequalities. Then solve each inequality.

x + 6 ≥ − 3 and x + 6 ≤ 3 Write related inequalities.

− 6 − 6 − 6 − 6 Subtract 6 from each side.

x ≥ − 9 and x ≤ − 3 Simplify.

The solution is x ≥ − 9 and x ≤ − 3.

15 12 9 6 3 0 3

b. Solve ∣ 2x − 5.5 ∣ ≥ −7.7.

The absolute value of an expression cannot be negative. The absolute value of an expression cannot be negative. So, So, ∣ 2x − 5.5 5.5 ∣ is nonnegative for all possible values of is nonnegative for all possible values of x.

The solution is all real numbers.

An absolute value inequality is an inequality that contains an absolute value expression. For example, ∣ x ∣ < 2 and ∣ x ∣ > 2 are absolute value inequalities.

∣ x ∣ < 2 ∣ x ∣ > 2

The distance between The distance between x and 0 is less than 2. x and 0 is greater than 2.

The graph of ∣ x ∣ < 2 The graph of ∣ x ∣ > 2 is x > − 2 and x < 2. is x < − 2 or x > 2.

You can solve absolute value inequalities by solving two related inequalities.

Key Vocabularyabsolute value inequality, p. 2

Solving Absolute Value Inequalities

To solve ∣ ax ∣ + b < c for c > 0, solve the related inequalities

ax + b > − c and ax + b < c.

To solve ∣ ax + b ∣ > c for c > 0, solve the related inequalities

ax + b < − c or ax + b > c.

In the inequalities above, you can replace < with ≤ and > with ≥ .

Do not assume there is no solution because one side is negative. Check the inequality symbol.

Common Error

3 2 1 0 1 2 3 3 2 1 0 1 2 3

Laurie’s Notes

T-2

Goal Today’s lesson is solving absolute value inequalities.

IntroductionConnect• It is recommended that this lesson be done after the lesson on solving

multi-step inequalities.• Yesterday: Students solved multi-step inequalities.• Today: Students will represent relationships in various contexts with

inequalities involving absolute value of linear expressions. Students will solve such inequalities and graph the solutions on a number line.

Motivate• Begin with a review of solving absolute value equations.

“When should you use an open circle and when should you use a closed circle when graphing the solution of an inequality?” Use an open circle when graphing an inequality with the symbols “<” and “>.” Use a closed circle when graphing an inequality with the symbols “≤” and “≥.”

Lesson Notes

• Tell students they can write an absolute value inequality as two related inequalities—similar to what they did with absolute value equations. “When should ‘and’ be used when writing an absolute value inequality as two related inequalities?” Use “and” when the absolute value expression is < or ≤ the constant value.

“When should ‘or’ be used when writing an absolute value inequality as two related inequalities?” Use “or” when the absolute value expression is > or ≥ the constant value.

• Note: An inequality with “and” can be written as a single inequality. For example, you can write x > 2 and x < 5 as 2 < x < 5.

Example 1• Work through part (a).

”Should the two related inequalities be joined by the word ‘and’ or ‘or’?” and

• Use the graph of the solution to reinforce the use of the word “and.” The solution of the absolute value inequality must be a solution to the fi rst inequality and the second inequality.

• Note: You can write x ≥ − 9 and x ≤ 5 as − 9 ≤ x ≤ 5. Show this to be true by graphing − 9 ≤ x ≤ 5.

• Work though part (b). Point out that regardless of the value of x, the absolute value expression will always be nonnegative. So, ∣ 2x − 5.5 ∣ will always be greater than − 7. The solution to the absolute value inequality is all real numbers.

• Common Misconception: As students solve absolute value inequalities involving negative numbers, point out that just because one side of the inequality is negative, it does not mean that there is no solution.

• Work through another example where the absolute value inequality has no solution. For instance, ∣ 7x − 9 ∣ < − 4 has no solution because regardless of the value of x, the absolute value expression is always nonnegative, which is not less than − 4.

Extra Example 1Solve the inequality. Graph the solution, if possible. a. ∣ x − 2 ∣ > 13 x < − 11 or x > 15

0 5 10 15 20510

11

15

b. ∣ x + 9 ∣ < − 1 no solution

c. ∣ 3x + 6 ∣ ≥ − 6 all real numbers

0 1 2 3123

Solving Absolute Value Inequalities 3

EXAMPLE Real-Life Appication33

You want to spend about $150 on a new cell phone. You are considering phones within $25 of $150. Write and solve an absolute value inequality to fi nd an acceptable price.

VARIABLE Let x represent the actual price of the cell phone.

WORDS Actualprice minus

amount youhave to spend

is less thanor equal to

the acceptabledifference.

INEQUALITY ∣ x − 150 ∣ ≤ 25

x − 150 ≥ − 25 and x − 150 ≤ 25 Write related inequalities.

+150 +150 +150 +150 Add 150 to each side.

x ≥ 125 and x ≤ 175 Simplify.

The prices you will pay must be at least $125 and at most $175.

Solve the inequality. Graph the solution, if possible.

1. ∣ x ∣ ≤ 3 2. ∣ x + 1 ∣ ≤ − 4 3. ∣ 2x + 1 ∣ ≤ 9

4. ∣ 6x + 1 ∣ > − 7 5. 2 ∣ 3x + 2 ∣ > 8 6. 2 ∣ − x + 1 ∣ − 8 ≥ 4

7. WHAT IF? In Example 3, you want to spend about $200 on a new cell phone. Find an acceptable price.

Exercises 3–12 and 14–23

EXAMPLE Solving an Absolute Value Inequality22

Solve 3 ∣ 4x − 2 ∣ > 18. Graph the solution.

3 ∣ 4x − 2 ∣ > 18 Write the inequality.

3 ∣ 4x − 2 ∣

— 3

> 18

— 3

Divide each side by 3.

∣ 4x − 2 ∣ > 6 Simplify.

Use ∣ 4x − 2 ∣ > 6 to write two related inequalities. Then solve each inequality.

4x − 2 < − 6 or 4x − 2 > 6 Write related inequalities.

+ 2 + 2 + 2 + 2 Add 2 to each side.

4x < − 4 or 4x > 8 Simplify.

4x

— 4

< − 4

— 4

or 4x

— 4

> 8

— 4

Divide each side by 4.

x < − 1 or x > 2 Simplify.

The solution is x < − 1 or x > 2.3 2 1 0 1 2 3

JohnHey, call me when you get...

KennySup?

Adamsoccer after school?

hen you get...hJohn h

Hey, call me wh

KennySup?

Adam hool?hhsoccer after sch

John

x

Th

Laurie’s Notes

T-3

Example 2 “What is the fi rst step in solving this absolute value inequality? Why?” Divide both sides by 3. So the absolute value expression is isolated on one side of the inequality before writing the absolute value inequality as two related inequalities.

• Use the graph of the solution to reinforce the use of the word “or.” The solution of the absolute value inequality must be a solution to the fi rst inequality or a solution to the second inequality.

Example 3• Explain: Develop the idea of acceptable difference. Tell students that this

amount is used to determine an acceptable price for the new cell phone. Basically, you are willing to pay anywhere from $125 to $175.

• Think-Pair-Share: Students should read the questions independently and then work with a partner to answer the questions. When they have answered the questions, the pair should compare their answers with another group and discuss any discrepancies.

• Explain: Show students how to graph a solution of all real numbers as done in Question 4. Provide a number line and shade the line in both directions.

Extra Example 2Solve the inequality. Graph the solution. a. 2 ∣ 4x − 2 ∣ ≤ 76 x ≥ − 9 and x ≤ 10

0 5 10 15 20510

9

15

b. ∣ x + 4.2 ∣ + 1.2 ≥ 7.6 x ≤ − 10.6 or x ≥ 2.2

0 4224681012

10.6 2.2

1. x ≥ − 3 and x ≤ 3

0 1 2 3123

2. no solution

3. x ≥ − 5 and x ≤ 4

0 2 4 6246

5

4. all real numbers

0 1 2 3123

5. x < − 2 or x > 2

— 3

0 1 2

23

31234

6. x ≤ − 5 or x ≥ 7

0 4 6 822468

5 7

7. ∣ x − 200 ∣ ≤ 25; x ≥ 175 and x ≤ 225; The price you will pay must be no less than $175 and no more than $225.

Extra Example 3

You want to spend about $70 on a pair of running shoes. You are considering shoes within $10 of $70. Write and solve an absolute value inequality to fi nd an acceptable price you are willing to pay. x − 70 ≤ 10; x ≥ 60 and x ≤ 80; The price you will pay is no less than $60 and no more than $80.

Closure• Exit Ticket: Explain when you should use related inequalities joined by the

word “and” and when you should use related inequalities joined by the word “or” when solving an absolute value inequality.

When ∣ ax + b ∣ is less than c and c is positive, use the word “and.”

When ∣ ax + b ∣ is greater than c and c is positive, use the word “or.”

4 Solving Absolute Value Inequalities

1. WRITING Compare and contrast solving absolute value equations and solving absolute value inequalities.

2. WHICH ONE DOESN’T BELONG Which does not belong with the other three? Explain your reasoning.

9 ∣ x + 4 ∣ ≤ 2

− x + ∣ 3 ∣ < 7

∣ x ∣ > 1

∣ x − 4 ∣ ≥ 6

9+(-6)=3

3+(-3)=

4+(-9)=

9+(-1)=

Solve the inequality. Graph the solution, if possible.

3. ∣ x ∣ ≥ 5 4. ∣ x ∣ ≤ 7

5. ∣ x ∣ < − 11 6. ∣ x ∣ > − 23

7. ∣ x − 2 ∣ ≤ 9 8. ∣ x + 4 ∣ > 12

9. ∣ x + 1.4 ∣ ≥ − 2.3 10. ∣ x − 1

— 4

∣ ≤ − 2

— 3

11. ∣ x + 8.5 ∣ < 3.9 12. ∣ x − 1

— 3

∣ > 1

— 2

13. ERROR ANALYSIS Describe and correct the error in solving the absolute value inequality.

Solve the inequality. Graph the solution, if possible.

14. ∣ x − 1 ∣ + 3 ≤ 20 15. ∣ x + 6 ∣ − 11 > 7

16. ∣ 2x − 4 ∣ ≥ 14 17. ∣ 3x + 3 ∣ < 15

18. ∣ 7x − 2 ∣ > − 17 19. ∣ x − 1 ∣ + 3 ≤ − 1

20. 2 ∣ x + 1 ∣ − 2 < 22 21. 3 ∣ x − 3 ∣ − 10 ≥ − 4

22. 3 ∣ − x − 1 ∣ + 9 ≥ 18 23. 2 ∣ − 4x − 4 ∣ − 8 < 8

24. SOCCER BALL You infl ate a new soccer ball to 7 pounds per square inch (psi). The instructions state that a pressure within 1 psi of 7 psi is acceptable. Write and solve an absolute value inequality to fi nd an acceptable pressure for the soccer ball.

11

22

Solve ∣ x − 1 ∣ < 3.

x − 1 < −3 or x − 1 > 3 x < −2 or x > 4

✗

Exercises

T-4

Assignment Guide and Homework Check

LevelLesson Assignment

HomeworkCheck

Basic 1, 2, 3–19 odd, 24 3, 7, 9, 17, 19, 24

Average 1–13, 15–23 odd, 24, 25 9, 15, 21, 25

Advanced 1, 2, 9–27 18, 21, 25, 26

Common Errors• Exercises 3 –12, 14–23 Students may try to solve an absolute value

inequality where the absolute value of an expression is said to be less than or equal to a negative number. For example, ∣ x + 3 ∣ ≤ − 6. Students may attempt to solve this inequality, but there is no solution.

• Exercises 3–12, 14–23 Students may incorrectly graph the inequality. For example, using an open dot when graphing an inequality involving the signs “≤” or “≥.”

• Exercises 22 and 23 Students may forget to reverse the direction of the inequality symbol when multiplying or dividing by a negative number.

3. x ≤ − 5 or x ≥ 5

0 5 10 1551015

4. x ≥ − 7 and x ≤ 7

0 5 10 155

7 7

1015

5. no solution

6. all real numbers

0 1 2 3123

7. x ≥ − 7 and x ≤ 11

0 5 10 155

7 11

1015

8. x < − 16 or x > 8

0 5 105

16 8

101520

9. all real numbers

0 1 2 3123

10. no solution

11. x > − 12.4 and x < − 4.6

2

12.4 4.6

468101214

12. x < − 1

— 6

or x > 5

— 6

016

16

26

36

46

56

13. See Additional Answers.

9+(-6)=3

3+(-3)=

4+(-9)=

9+(-1)=

1. Answer should include, but is not limited to: Absolute value equations and inequalities are both solved by writing each as two related equations or inequalities. Related inequalities are joined by “and” or “or” whereas equations are not.

2. − x + ∣ 3 ∣ < 7; This is not an absolute value inequality.

Solving Absolute Value Inequalities 5

Evaluate the expression when x = −2 and y = 3. (Skills Review Handbook)

28. x + 7y 29. y 2 − 3.4

30. 2x 2 − y 31. 1

— 2

xy 3

32. MULTIPLE CHOICE Which expression is not equivalent to 82? (Skills Review Handbook)

○A √—

64 ○B 8 × 8

○C 43 ○D 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8

25. SONG LENGTH An artist is recording a song. The goal is to make the song 3.5 minutes long. The producer gave the artist permission to be within 30 seconds of this goal.

a. Write and solve an absolute value inequality to fi nd an acceptable song length (in seconds).

b. List 3 acceptable song lengths. Check your solutions.

26. SNACKS A company sells 56 ounce bags of trail mix. Packaging guidelines state that a weight within 0.8 ounce of 56 ounces is acceptable.

a. Write and solve an absolute value inequality to fi nd an acceptable weight for a bag of trail mix.

b. Suppose the packaging guidelines state that the 56-ounce bag must weigh between 55.2 and 56.8 ounces. Write and solve an absolute value inequality to fi nd an acceptable weight for a bag of trail mix.

c. How do the inequalities from parts (a) and (b) differ? Explain your reasoning.

27. What is the solution to the inequality when c < 0? Explain your reasoning.

a. ∣ ax + b ∣ < c

b. ∣ ax + b ∣ > c

2

c.

T-5

Common Errors• Exercises 24–26 Students may incorrectly interpret which direction

the inequality symbol should point when setting up an absolute value inequality from a word problem.

• Exercises 24–26 Students may not understand which English words mean that a data value should or should not be included in the answer. For example, this will cause students to question whether to use > or ≥.

14. x ≥ − 16 and x ≤ 18

0 10 20

18

301020

16

30

15. x < − 24 or x > 12

0 10 20

12

301020

24

30

16. x ≤ − 5 or x ≥ 9

0 4 6 8 102246

5 9

17. x > − 6 and x < 4

0 4 6 822468

18. all real numbers

0 1 2 3123

19. no solution

20. x > − 13 and x < 11

0 5 10 155

13 11

1015

21. x ≤ 1 or x ≥ 5

0 1 2 3 54 61

22. x ≤ − 4 or x ≥ 2

0 4 6 822468

23. x > − 3 and x < 1

0 1 2 31234

24. ∣ x − 7 ∣ ≤ 1; x ≥ 6 and x ≤ 8

25. a. ∣ x − 210 ∣ ≤ 30; x ≥ 180and x ≤ 240

b. Sample answers: 190, 200, or 230 seconds

26–27. See Additional Answers.

9+(-6)=3

3+(-3)=

4+(-9)=

9+(-1)=

28. 19 29. 5.6

30. 5 31. − 27

32. A