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Name Class Date
Prentice Hall Algebra 2 ã
Teaching Resources
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
59
1-6
Reteaching
Absolute Value Equations and Inequalities
Solving absolute value equations require solving two equations separately. Recall that for a real number
x
,
u
x
u
is the distance from zero to
x
on the number line. Te equation
u
x
u
5
p
means that either
x
5
p
or
x
52
p
because both are
p
units from 0.
Problem
What is the solution set for the equation
u
5
x
1
1
u
2
3
5
4?Te rst step in solving an absolute value equation is to isolate the absolute value on one side of the equal sign.
u
5
x
1
1
u
2
3
5
4
u
5
x
1
1
u
2
3
1
3
5
4
1
3
Add 3 to each side.
u
5
x
1
1
u
5
7
Simplify.
Next, rewrite the absolute value as two equations and solve each of them separately.5
x
1
1
5
7
or
5
x
1
1
52
7
Deﬁnition of absolute value
5
x
5
6
or
5
x
52
8
Addition Property of Equality
x
5
65
or
x
52
85
Division Property of Equality
Notice that the same operations are performed in the same order on each of the two equations. However, do not try to “simplify” the process by solving a single equation. Tis leads to errors.Te solutions are
x
5
65
or
x
52
85
. Check each solution in the srcinal equation:
Check
P
5
?
65
1
1
P
2
3
5
4
P
5
?
Q
2
85
R
1
1
P
2
3
5
4
u
6
1
1
u
2
3
5
4
u
2
8
1
1
u
2
3
5
4 4
5
4
✓
4
5
4
✓
Exercises
Solve each absolute value equation. Check your work.
1.
u
2
x
2
3
u
2
4
5
3
2.
u
3
x
2
6
u
1
1
5
13
Name Class Date
Prentice Hall Algebra 2 ã
Teaching Resources
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
60
1-6
Reteaching
(continued)
Absolute Value Equations and Inequalities
o solve an absolute value inequality, keep in mind that
u
x
u
is the distance from zero to
x
on the number line. So, if
u
x
u
,
p
, then
x
is less than
p
units from 0, so
u
x
u
,
p
1
2
p
,
x
,
p
. And, if
u
x
u
.
p
, then
x
is greater than
p
units from 0, so
u
x
u
.
p
1
x
,2
p
or
x
.
p
.In this case, we need to rewrite the absolute value inequality as two separate inequalities. Do not try to combine them into one inequality.
Problem
What is the solution set for the inequality
u
2
x
1
3
u
.
11?Because the inequality is
.
, use
u
x
u
.
p
1
x
,2
p
or
x
.
p
.Begin by rewriting the absolute value as two equations and solve each of them separately. 2
x
1
3
,
2
11 or 2
x
1
3
.
11
Rewrite as a compound inequality.
2
x
,
2
14 or 2
x
.
8
Subtract 3 from each side.
x
.
2
7 or
x
.
4
Divide each side by 2.
Te solution set is
x
,
2
7 or
x
.
4.
Exercises
Complete the steps to solve the inequality
P
x
2
2
4
P
K
3.
3.
#
x
2
2
4
#
Rewrite as a compound inequality.
4.
u
#
x
2
#
u
Add
u
to each part.
5.
#
x
#
Multiply each part by .
6.
What is the solution?

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