Solving Absolute Value Inequalities

Solving Absolute Value Inequalities

Key Vocabulary

absolute value inequality, p. 2

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 x and 0 is less than 2.

The distance between x and 0 is greater than 2.

3 2 1 0 1 2 3

The graph of x < 2

is x > -2 and x < 2.

3 2 1 0 1 2 3

The graph of x > 2

is x < -2 or x > 2.

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

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 .

EXAMPLE 1 Solving Absolute Value Inequalities

a. Solve x + 6 3. Graph the solution. Use x + 6 3 to write two related inequalities. Then solve

each inequality.

x + 6 -3 and x + 6 3

-6 -6

-6 -6

x -9 and

x -3

Write related inequalities. Subtract 6 from each side. Simplify.

The solution is x -9 and x -3.

Common Error

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

15 12 9 6 3 0 3

b. Solve 2x - 5.5 -7.

The absolute value of an expression cannot be negative.

So, 2x - 5.5 is nonnegative for all possible values of x.

The solution is all real numbers.

2

Solving Absolute Value Inequalities

Laurie's Notes

Introduction

Connect

? 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 "." Use a closed circle when graphing an inequality with the symbols "" and "."

Lesson Notes

Goal Today's lesson is solving

absolute value inequalities.

? 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 first 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 1 Solve the inequality. Graph the solution, if possible.

a. x - 2 > 13

x < -11 or x > 15

11 15 10 5 0 5 10 15 20

b. x + 9 < -1

no solution

c. 3x + 6 -6

all real numbers

3 2 1 0 1 2 3

T-2

EXAMPLE 2 Solving an Absolute Value Inequality

Solve 3 4x - 2 > 18. Graph the solution.

3 4x - 2 > 18

Write the inequality.

3 4x - 2

--

>

18 --

3

3

4x - 2 > 6

Divide each side by 3. 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 --

or

4 4

x < -1 or

4x --

>

8 --

4 4

x > 2

Divide each side by 4. Simplify.

The solution is x < -1 or x > 2.

3 2 1 0 1 2 3

EXAMPLE 3 Real-Life Appication

John Hey,

call

me

when

you

get...

Kenny Sup?

Asodcacmer after school?

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 find an acceptable price.

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

Actual

amount you is less than the acceptable

WORDS price minus have to spend or equal to difference.

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.

Exercises 3?12 and 14?23

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.

Solving Absolute Value Inequalities

3

Extra Example 2

Solve the inequality. Graph the solution.

a. 2 4x - 2 76

x -9 and x 10

9

15 10 5 0 5 10 15 20

b. x + 4.2 + 1.2 7.6

x -10.6 or x 2.2

10.6

2.2

1210 8 6 4 2 0 2 4

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 find 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.

1. x -3 and x 3

3 2 1 0 1 2 3

2. no solution 3. x -5 and x 4

5 6 4 2 0 2 4 6

4. all real numbers

3 2 1 0 1 2 3

5. x < -2 or x > --2 3

2 3

4 3 2 1 0 1 2 3

6. x -5 or x 7

5

7

8 6 4 2 0 2 4 6 8

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.

Laurie's Notes

Example 2

"What is the first 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 first 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.

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."

T-3

Exercises

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

93++4(-+(6-9(3)-=+)9=3()-=1)=

Solve the inequality. Graph the solution, if possible.

1 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 ---2

4

3

11. x + 8.5 < 3.9

12. x - --1 > --1 3 2

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

Solve x - 1 < 3.

x - 1 < -3 or x - 1 > 3

x < -2 or

x > 4

Solve the inequality. Graph the solution, if possible.

2 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 inflate 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 find an acceptable pressure for the soccer ball.

4

Solving Absolute Value Inequalities

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