Download 2-6 Solve Absolute

2-6 Solve Absolute-Value Equations
Name
Date
Check: 3|x ⫹ 1| ⫽ 12
?
?
3|3 ⫹ 1| ⫽ 12 and 3|5 + 1| ⫽ 12
?
?
3|4| ⫽ 12
3|4| ⫽ 12
?
?
3(4) ⫽ 12
3(4) ⫽ 12
Solve: 3|x ⫹ 1| ⫽ 12
3| x ⫹ 1 | 12
⫽
3
3
Divide by 3 to isolate the
absolute-value expression.
|x ⫹ 1| ⫽ 4
Consider two cases.
x⫹1⫽4
or
x ⫹ 1 ⫽ ⫺4
x⫹1⫺1⫽4⫺1
x⫹1⫺1⫽⫺4⫺1
Solve each
12 12 True
12 12 True
equation.
x⫽3
x ⫽ ⫺5
Solution set: {⫺5, 3}
Find the solution for each equation. Then check your solutions.
1. |b| ⫽ 2
2. | d | ⫽ 11
b 2 or b –2
Check: | b | 2
?
3. |r | ⫽ 9.1
4. |v| ⫽ 15.4
?
| 2 | 2 AND |–2| 2
2 2 True 2 2 True
{2, 2}
{11, 11}
5
4
5. | p| ⫽ 8
6. |w| ⫽ 5
{58, 58}
Copyright © by William H. Sadlier, Inc. All rights reserved.
m – 5 34
m – 5 34
{29, 39}
13. ⫺| 2u| ⫽ ⫺16
u8
u 8
{8, 8}
1
3
17. | g ⫹ 2 | ⫽ 4
2 3
1
g g
4 4
4
g
2
3
5
g
4
4
4
5
1
4, 4
{
}
t – 14 47
t – 14 47
11. |7x | ⫽ 28
14. ⫺|7w| ⫽ ⫺98
{14, 14}
3
1
18. | h ⫹ 8 | ⫽ 2
1
3 4
h h
8
8 8
7
3
4
h h
8
8
8
7
1
8, 8
{
x 4
x4
{4, 4}
15. |v ⫹ 1.2| ⫽ 3.7
v 1.2 3.7
w 14
v 1.2 3.7
w 14
}
19. |11w | ⫽ ⫺88
| 11w | cannot be
less than 0
⭋
k 31
k 13
{31, 13}
5y 95
5y 95
y 19
y 19
{ 19, 19}
16. |t ⫹ 7.3| ⫽ 4.8
v 4.9 t 7.3 4.8
v 2.5
t 7.3 4.8
{4.9, 2.5}
Lesson 2-6, pages 58–59.
k 9 22
k 9 22
12. |5y| ⫽ 95
7x 28
7x 28
{33, 61}
7w 98
7w 98
f 20
f6
{20, 6}
t 61
t 33
{15.4, 15.4}
8. |k ⫹ 9| ⫽ 22
f 7 13
f 7 13
10. |t ⫺ 14| ⫽ 47
m 39
m 29
2u 16
2u 16
7. | f ⫹ 7| ⫽ 13
{45, 45}
9. |m ⫺ 5| ⫽ 34
{9.1, 9.1}
t 12.1
t 2.5
{12.1, 2.5}
20. ⫺|2n| ⫽16
| 2n| cannot be less
than 0
⭋
Chapter 2
49
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Find the solution set for each equation. Then check your solutions.
21. |k ⫹ 5| ⫺ 2 ⫽ 23
|k 5| 2 23
|k 5| 25
k 5 25 or k 5 25
k 20
k 30
{30, 20}
22. | j ⫹ 7| ⫺ 4 ⫽ 15
?
Check: |30 5| 2 23
23 23
True
?
|20 5| 2 23
23 23
True
24. |q ⫺ 2| ⫹ 11 ⫽ 11
| j 7 | 19
j 26 or j 12
| n 1| 5
|n 1| cannot be less than 0.
{ 4}
25. 3| n ⫹ 1| ⫽ 18
26. 4| m ⫹ 3| ⫽ 40
| m 3 | 10
m 7 or m 13
| n 1| 6
n 5 or n 7
{2}
27. ⫺5| n ⫹ 1| ⫽ 25
| j 4 | 0; j 4
{ 26, 12}
| q 2| 0;q 2
⭋
23. | j – 4| ⫹ 8 ⫽ 8
{13, 7}
{7, 5}
28. 2| a ⫹ 23| ⫽ ⫺12
29. 6|z ⫺ 2| ⫹ 4 ⫽ 40
| a 23 | 6
| a 23| cannot be less than 0.
⭋
6 | z 2 | 36
|z 2| 6
z 4 or 8
{4, 8}
30. Twelve more than the absolute value of a
number decreased by 7 is 19. What are the
possible values for the number?
Let x the number; |x 7| 12 19;
|x 7| 7; x 14 or x 0
The possible values are 0 and 14.
31. Kyle wants to score within 4.5% of 94% on
his next test. What are the maximum and
minimum scores he is aiming for?
Let s Kyle’s score. |s 94| 4.5; s 98.5
or s 89.5; Kyle’s maximum score is 98.5%
and his minimum score is 89.5%.
32. Why does the equation |x ⫺ 4| ⫽ ⫺2 have no solution? (Hint: Use the
concept that the absolute value of two numbers is the distance between them.)
Possible answer: Distance is never negative, so the distance between x and
4 cannot be negative. Because the distance between two numbers cannot be
negative, this also means that |x 4| cannot be negative.
50
Chapter 2
Copyright © by William H. Sadlier, Inc. All rights reserved.
Write and solve an equation for each problem. Check your solutions.