Download 2 Solve an Absolute-Value Inequality 3 Solve an

STUDY TIP
The expression inside
the absolute-value
symbols can be
positive or negative.
When you rewrite the
expression for the
negative value, reverse
the inequality.
Solve an Absolute-Value Inequality
2
EXAMPLE
Student Help
Solve ⏐x 4⏐ < 3. Then graph the solution.
Solution The solution consists of all numbers x whose distance from 4 is
less than 3. The inequality involves < so the related inequalities are connected
by and.
⏐x 4⏐ < 3
Write original inequality.
x4<3
and
x 4 > 3
x44<34
and
x 4 4 > 3 4
x<7
ANSWER 䊳
and
Write related inequalities.
Add 4 to each side.
x>1
Simplify.
The solution is all real numbers greater than 1 and less than 7.
This can be written 1 < x < 7. The graph of the solution is
shown below.
1
0
1
2
3
4
5
6
7
8
9
Check the solution.
Student Help
STUDY TIP
When you check your
solution, choose values
that make substitution
simple. In Example 3
you might choose 10,
5, and 0.
Solve an Absolute-Value Inequality
3
EXAMPLE
Solve ⏐x 5⏐ ≥ 2. Then graph the solution.
Solution The solution consists of all numbers x whose distance from 5 is
greater than or equal to 2. The inequality involves ≥ so the related inequalities
are connected by or.
⏐x 5⏐ ≥ 2
Write original inequality.
x5≥2
or
x 5 ≤ 2
Write related inequalities.
x55≥25
or
x 5 5 ≤ 2 5
Subtract 5 from each side.
x ≥ 3
ANSWER 䊳
x ≤ 7
or
Simplify.
The solution is all real numbers greater than or equal to 3 or less
than or equal to 7. This can be written x ≤ 7 or x ≥ 3.
The graph of the solution is shown below.
10
9
8
7
6
5
4
3
2
1
0
Check the solution.
Solve an Absolute-Value Inequality
Solve the absolute-value inequality.
362
Chapter 6
1. ⏐x⏐ ≤ 6
2. ⏐x 2⏐ < 5
3. ⏐x 1⏐ ≤ 4
4. ⏐3x⏐ > 9
5. ⏐x 2⏐ ≥ 7
6. ⏐x 3⏐ > 12
Solving and Graphing Linear Inequalities