Warm Up
Solve the inequality and graph the solutions.
1.
x+7<4
3.
5 + 2x > 1
2.
14x > 28
==========================================================================
2-7: Solving Absolute-Value Inequalities
Objectives:
Solve compound inequalities in one variable involving absolute-value expressions.
When an inequality contains an absolute-value expression, it can be written as a compound inequality. The
inequality |x| < 5 describes all real numbers whose distance from 0 is less than 5 units. The solutions are all
numbers between –5 and 5, so |x|< 5 can be rewritten as –5 < x < 5, or as x > –5 AND x < 5.
1
Helpful Hint
Just as you do when solving absolute-value equations, you first isolate the absolute-value
expression(if needed) when solving absolute-value inequalities.
Example 1:
A. Solve the inequality and graph the solutions:
|x| – 3 < –1
B. Solve the inequality and graph the solutions:
|x – 1| ≤ 2
C.I.O.-Example 1:
a. Solve the inequality and graph the solutions:
2|x| ≤ 6
b. Solve the inequality and graph the solutions:
|x + 3|– 4.5 ≤ 7.5
2
The inequality |x| > 5 describes all real numbers whose distance from 0 is greater than 5 units. The solutions are
all numbers less than –5 or greater than 5. The inequality |x| > 5 can be rewritten as the compound inequality
x < –5 OR x > 5.
Example 2:
A.
Solve the inequality and graph the solutions.
B.
|x| + 14 ≥ 19
Solve the inequality and graph the solutions.
3 + |x + 2| > 5
C.I.O.-Example 2:
a.
Solve the inequality and graph the solutions.
b.
Solve the inequality and graph the solutions.
|x| + 10 ≥ 12
3
Additional Example 3:
A pediatrician recommends that a baby’s bath water be 95°F, but it is acceptable for the temperature to vary
from this amount by as much as 3°F. Write and solve an absolute-value inequality to find the range of
acceptable temperatures. Graph the solutions.
Let t represent the actual water temperature.
C.I.O.-Example 3:
A dry-chemical fire extinguisher should be pressurized to 125 psi, but it is acceptable for the pressure to differ
from this value by at most 75 psi. Write and solve an absolute-value inequality to find the range of acceptable
pressures. Graph the solution.
Let p represent the desired pressure.
4
When solving an absolute-value inequality, you may get a statement that is true for all values of the variable. In
this case, all real numbers are solutions of the original inequality. If you get a false statement when solving an
absolute-value inequality, the original inequality has no solutions.
Example 4: Special Cases of Absolute-Value Inequalities
A. Solve the inequality:
|x + 4| – 5 > – 8
B. Solve the inequality:
|x – 2| + 9 < 7
Remember!
An absolute value represents a distance, and distance cannot be less than 0.
C.I.O.-Example 4:
a. Solve the inequality:
|x| – 9 ≥ –11
b. Solve the inequality:
4|x – 3.5| ≤ –8
5
Lesson Quiz: Part I
Solve each inequality and graph the solutions.
1.
3|x| > 15
2.
|x + 3| + 1 < 3
3. A number, n, is no more than 7 units away from 5. Write and solve an inequality to show the range of
possible values for n.
Lesson Quiz: Part II
Solve each inequality.
4.
|3x| + 1 < 1
5.
|x + 2| – 3 ≥ – 6
p. 145: 21-31 odd, 32, 33-37 odd, 39-49, 51
Even #’s answer:
32)
40) never
6