4
EXAMPLE
Solve a Multi-Step Inequality
Solve ⏐x 1⏐ 3 ≥ 2.
Solution
First isolate the absolute-value expression on one side of the inequality.
⏐x 1⏐ 3 ≥ 2
⏐x 1⏐ 3 3 ≥ 2 3
⏐x 1⏐ ≥ 5
Write original inequality.
Add 3 to each side.
Simplify.
The inequality involves ≥ so the related inequalities are connected by or.
⏐x 1⏐ ≥ 5
x1≥5
or
x11≥51
x≥4
ANSWER 䊳
Write simplified inequality.
x 1 ≤ 5
or x 1 1 ≤ 5 1
x ≤ 6
or
Write related inequalities.
Subtract 1 from each side.
Simplify.
The solution is all real numbers greater than or equal to 4 or less
than or equal to 6. This can be written x ≤ 6 or x ≥ 4.
Solve a Multi-Step Inequality
7. Solve the inequality ⏐3x 2⏐ > 4.
Student Help
STUDY TIP
Compare Example 5 to
Example 5 on page 350.
Together the examples
show the connection
between absolutevalue inequalities and
compound inequalities.
5
EXAMPLE
Use an Absolute-Value Inequality
A baseball is hit straight up with an initial velocity of 64 feet per
second. Its speed s (in ft/sec) after t seconds is given by s ⏐32t 64⏐.
Find the values of t for which s is greater than 32 feet per second.
BASEBALL
Solution Solve ⏐32t 64⏐ > 32.
The inequality involves > so the related inequalities are connected by or.
⏐32t 64⏐ > 32
32t 64 > 32
or 32t 64 < 32
32t > 32 or
t<1
ANSWER 䊳
Write original inequality.
32t < 96
or
t>3
Write related inequalities.
Subtract 64 from each side.
Divide by 32 and reverse
the inequalities.
The speed is greater than 32 ft/sec when t is less than 1 second or
greater than 3 seconds. This can be written t < 1 or t > 3.
Use an Absolute-Value Inequality
8. In Example 5 find the values of t for which s is greater than 48 ft/sec.
6.7
Solving Absolute-Value Inequalities
363