Absolute Value Equations
and Inequalities
1-6
Vocabulary
Review
1. Circle the solution of 3x 1 8 5 24.
x 5 29
x 5 24
x 5 43
x54
Vocabulary Builder
absolute value
absolute value (noun)
AB
suh loot VAL yoo
2
Definition: The absolute value of a real number x is its
distance from zero on the number line.
2 1
Main Idea: If x is positive, then u x u 5 x . If x is negative,
then u x u 5 2x .
2
0
1
2
2 and 2 are
2 units from 0
Use Your Vocabulary
Graph each absolute value on the number line.
2. u 3 u
3. u 22 u
6
5
4
3
2
4. 2u 1 u
1
0
1
2
5. 2u 22 u
3
4
5
Complete each graph.
6. u x u 5 4
4
Chapter 1
7. u x u 5 5
0
4
4
22
0
4
6
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Examples: u 5 u 5 5, u 25 u 5 5
Problem 1 Solving an Absolute Value Equation
Got It? What is the solution of »3x 1 2… 5 4? Graph the solution.
8. Write as two equations.
or
9. Complete the steps to find x.
Write the two equations from Exercise 8.
Subtract 2 from each side.
Divide each side by 3.
10. Graph the solution.
6
5
4
3
2
1
0
1
2
3
4
5
6
Problem 2 Solving a Multi-Step Absolute Value Equation
Got It? What is the solution of 2»x 1 9… 1 3 5 7? Graph the solution.
11. Complete the steps to find x.
2u x 1 9 u 1 3 5 7
Write the original equation.
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
2u x 1 9 u 5
Subtract 3 from each side.
ux 1 9u 5
Divide each side by 2.
12. Write as two equations.
or
13. Solve for x.
Write the two equations from Exercise 12.
Subtract 9 from each side.
14. Graph the solution.
12
10
8
6
4
2
0
2
23
4
6
8
10
12
Lesson 1-6
Problem 3
Solving for Extraneous Solutions
Got It? What is the solution of »5x 2 2… 5 7x 1 14? Check for extraneous solutions.
15. Solving the equation yields x 5 21 and x 5 28. Substitute each value into the
original equation.
u5 ?
2 2u 5 7 ?
u
1 14
u5 ?
2 2u 5 7 ?
u 5
u
1 14
u 5
16. Which, if any, solutions are extraneous? Explain.
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Problem 4
Solving the Absolute Value Inequality »A… R b
Got It? What is the solution of »3x 2 4… K 8? Graph the solution.
17. The inequality is solved below. Use the justifications to complete each step.
#8
Write the compound inequality.
#
#
Add 4 to each part.
#
#
Divide each part by 3.
28 #
18. The endpoints of the graph are
and
.
19. Now graph the solution.
10
Problem 5
8
6
4
2
0
2
4
6
8
10
Solving the Absolute Value Inequality »A… L b
Got It? What is the solution of »5x 1 10… S 15? Graph the solution.
20. Write the absolute value inequality as a compound inequality.
5x 1 10 , 215
or
21. Solve each inequality for x.
22. Now graph the solution.
10
Chapter 1
8
6
4
2
0
2
24
4
6
8
10
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
The endpoints are indicated on the graph with open / closed dots.
Problem 6 Using an Absolute Value Inequality
Got It? Suppose the smallest allowable height of a racecar is 52 in. and the
desirable height is 52.5 in. What absolute value inequality describes heights
of a racecar within an indicated tolerance?
23. Find the difference between the desirable height and the smallest allowable height.
52.5 2
5
24. Circle the expression that models the difference between the desirable height and
the actual height h.
52.0 2 h
52.5 2 h
25. Write a compound inequality.
20.5 #
# 0.5
26. Rewrite the compound inequality as an absolute value inequality.
u
u#
Lesson Check • Do you UNDERSTAND?
Reasoning When is the absolute value of a number equal to itself?
27. The absolute value of 0 is equal to 9 .
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
28. When a number is negative, its absolute value is equal to 9 .
29. When a number is positive, its absolute value is equal to 9 .
30. Now answer the question.
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Math Success
Check off the vocabulary words that you understand.
absolute value
graph
inequality
equation
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Lesson 1-6