3-6 Solve Absolute-Value Inequalities
Name
Date
Solve: 2|c 6| 1 11
2|c 6| 1 1 11 1
2|c 6| 12
2
2
|c 6| 6
c 6 6 OR c 6 6
6 6
6 6
c0
Graph:
{c | c 0 OR c 12} OR (, 0] [12, )
Isolate the absolute-value
expression.
Use the Addition
Property of Inequality.
2
0
2
4
6
8 10 12 14
Check: c 1 and 20
?
2|(1) 6| 1 11 OR 2|20 6| 1 11
?
?
2|7| 1 11
2|14| 1 11
?
?
2(7) 1 11
28 1 11
13 11
27 11
True
True
Use the Division Property
of Inequality.
Rewrite this statement as a
disjunction.
Use the Addition Property
of Inequality.
c 12
OR
Remember: If an absolute-value inequality involves the “less than” symbol,
rewrite the inequality as a conjunction.
Solve the inequality. Write the answer in set-builder and
interval notation. Check your work. Answer checks will vary. Check students’ work.
1. |a| 9 12
2. | d | 6 7
Copyright © by William H. Sadlier, Inc. All rights reserved.
| a | 9 9 , 12 9
|a| , 3
a . 3 AND a , 3
{a | 3 , a , 3}; (3, 3)
Check: a 0
?
| 0 | 9 # 12
9 # 12 True
|d| , 1
d . 1 AND d , 1
{ d | 1 , d , 1}; (1, 1)
Check: d 0
?
|0| 6 , 7
6 , 7 TRUE
4. | f | 7 11
|w| . 5
w . 5 OR w , 5
{ w | w , 5 or w . 5}; (`, 5) (5, `)
Check: w 10
?
| 10 | 2 . 3
8 . 3 TRUE
5. 6| x| 18
| f | . 18
f , 18 OR f . 18
{ f | f , 18 or f . 18}; (`, 18) (18, `)
Check: f 20
?
| 20 | 7 . 11
13 . 11 TRUE
7. 8|y | 6 42
3. | w| 2 3
6. 5| t | 30
|x| # 3
x $ 3 AND x # 3
{ x | 3 # x # 3}; [3, 3]
Check: x 0
?
6 | 0 | # 18
0 # 18 TRUE
8. 3| z| 18 27
9. 4| b | 4 16
8| y | $ 48; | y | $ 6
3| z | . 9; | z | . 3
y # 6 or y $ 6
z , 3 or z . 3
{ y | y # 6 or y $ 6}; (`, 6] [6, `) { z | z , 3 or z . 3}; (`, 3) (3, `)
Check: y 10
Check: z 4
?
8 | 10| 6 $ 42
74 $ 42 TRUE
|t| # 6
t $ 6 AND t # 6
{ t | 6 # x # 6}; [6, 6]
Check: t 1
?
5 |1| # 30
5 # 30 TRUE
?
3 | 4| 18 . 27
30 . 27 TRUE
Lesson 3-6, pages 84–85.
4| b | . 20; | b | , 5
b . 5 AND b , 5
{ b | 5 , b , 5}; (5, 5)
Check: b 0
?
4 | 0 | 4 . 16
4 . 16 TRUE
Chapter 3
75
For More Practice Go To:
Solve the inequality. On a separate sheet of paper, graph and check
the solution set. Answers may vary. Check students’ answers.
2
11. | m 4| 6 9
| q 2 5 | 13 # 14
| q 2 5| # 1
q 2 5 $ 21 AND q 2 5 # 1
q $ 4 AND q # 6
4#q#6
12. 3 |a | 5
| m 2 4| # 3
m 2 4 $ 23 AND m 2 4 # 3
m $ 1 AND m # 7
1#m#7
3
4
13. 4 | b | 3
1
15
2
15
15
a # 2 OR a $
2
2
|a| $
1
14. |2x 3| 4 5 6 5
| 2 x 1 3| . 11
2x 1 3 , 211 OR 2x 1 3 . 11
2x , 214 OR
2x . 8
x , 27 OR
x.4
|b| $ 4
b # 24 OR b $ 4
16. | 5a 3| 9 7
17. |4g 9| 3 7
| 5 a 1 3 | , 16
5a 1 3 . 216 AND 5a 1 3 , 16
5a . 219 AND
5a , 13
13
19
a,
a . 2 AND
5
5
13
19
2 ,a,
5
5
|2 x 2 4 | . 8
2x , 24 OR 2x . 12
x , 22 OR x . 6
18. 4 |2h 1| 7
|4g 2 9 | , 10
4g 2 9 . 210 AND 4g 2 9 , 10
4g . 21 AND
4g , 19
19
1
g,
g . 2 AND
4
4
1
19
2 ,g,
4
4
19. 4 |3k 2| 9
20. 1.7 |ᐉ 3| 2.8
|3k 2 2| # 13
3k 2 2 $ –13 AND 3k 2 2 # 13
3k $ 211 AND
3k # 15
11
k#5
k $ 2 AND
3
11
2 #k#5
3
|2h 2 1| # 11
2h 2 1 $ 211 AND 2h 2 1 # 11
2h $ 210 AND
2h # 12
h $ 25 AND
h#6
25 # h # 6
21. 3.5 |m 2| 5.7
| 2 3 | $ 1.1
2 3 # 21.1 OR 2 3 $ 1.1
# 1.9 OR
$ 4.1
# 1.9 or $ 4.1
22. What integers are the solutions to
| x 4| 8 AND |3x| 12?
5
15. |2x 4| 3 6 4 6
|m 2 2| $ 2.2
m 2 2 # 22.2 OR m 2 2 $ 2.2
m # 20.2 OR
m $ 4.2
m # 20.2 or m $ 4.2
23. What integers are the solutions to
| x 1| 4 OR |x 1| 3?
The solution to | x 2 4 | , 8 is 24 , x , 12
The solution to |23x | . 12 is x , 24 OR x . 4.
There are no integers in 24 , x , 12 AND x , 24.
The integers in 24 , x , 12 AND x . 4 are
5, 6, 7, 8, 9, 10, 11.
The solution to | x 1 1| , 4 is 25 , x , 3, with
integer solutions 24, 23, 22, 21, 0, 1, 2. The
solution to |2x 2 1| , 3 is 24 , x , 2, with integer
solutions 23, 22, 21, 0, 1. The union of the disjoint
sets {x |25 , x , 3} OR {x |24 , x , 2} is
{x |25 , x , 3}, with integers 24, 23, 22, 21, 0, 1, 2.
Write rational or irrational for each number.
2
24. 3 rational
25.
5 irrational
26.
27. irrational
100 rational
Find the value of each expression.
28.
32
72
110
8
1
49
29. 19 21 (2)[8(3) 4]
42
30.
2 13
35
18
1 15
Solve.
31. 3x 19 21
2
x
3
76
Chapter 3
32. 5c 17 2c 14
31
c
3
33. 3(2b 4) 7 (b 8)
b
11
7
Copyright © by William H. Sadlier, Inc. All rights reserved.
10. |q 5| 13 14
Answers for Algebra I, Practice Book Lesson 3-6, page 76.
10.
11.
12.
28 26 24 22 0 2 4 6 8
13.
14.
28 26 24 22 0 2 4 6 8
15.
28 26 24 22 0 2 4 6 8
16.
17.
25 24 23 22 21 0 1 2 3
26 24 22 0 2 4 6 8 10
18.
0
19.
1
2
3
4
5
6
28 26 24 22 0 2 4 6 8
20.
24 23 22 0 1 2 3 4 5
1.5
2.0
21.
1
28 26 24 22 0 2 4 6 8
0 1 2 3 4 5 6 7 8
0
76A Chapter 3
1
2
3
4
5
2.5
3.0
3.5
4.0