Math Help for Section 3.7, Page 155
Solving
Inequalities Involving Absolute Value
To see how to solve inequalities involving absolute value, consider the
following
comparisons.
| $|x|=2$ | $|x|<2$ | $|x|<2$ | |||
| $x=-2$ and $x=2$ | $-2<x<2$ | $x<-2$ or $x>2$ | |||
 |
|||||
These comparisons illustrate the rules in the box on page 89 for
solving inequalities
involving
absolute value.
Example
6: Tip
The solution set in set notation is $\{x|\,3<x<7\}$.
Study
Tip
To get an indication of the validity of a solution set of an
absolute value inequality, you should check
values
in the solution set and outside of the solution set. In Example 6 on
page 89 you
can check that $x=4$ is in the solution set and that $x=2$ and $x=8$
are not in the solution set.
Example
6: Check
The solution was found to be $3<x<7$. As stated above,
check that
$x=4$
satisfies the inequality and that $x=2$ and $x=8$ do not.
| $x=4$: | |
| $\eqalign{|x-5|< & 2 &{\small\color{red}\quad\quad \text{Write original inequality.}} \cr |{\color{red}4}-5| \overset{?}{<}& 2 &{\small\color{red}\quad\quad \text{Substitute 4 for }x.} \cr |-1|\overset{?}{<}& 2 &{\small\color{red}\quad\quad \text{Subtract.}} \cr 1<& 2 &{\small\color{red}\quad\quad \text{Solution checks. }\checkmark} } $ |
|
| $x=2$: | |
| $\eqalign{|x-5|< & 2 &{\small\color{red}\quad\quad \text{Write original inequality.}} \cr |{\color{red}2}-5| \overset{?}{<}& 2 &{\small\color{red}\quad\quad \text{Substitute 2 for }x.} \cr |-3|\overset{?}{<}& 2 &{\small\color{red}\quad\quad \text{Subtract.}} \cr 3\not<& 2 &{\small\color{red}\quad\quad \text{Solution does not check. ✗}} } $ |
|
| $x=8$: | |
| $\eqalign{|x-5|< & 2 &{\small\color{red}\quad\quad \text{Write original inequality.}} \cr |{\color{red}8}-5| \overset{?}{<}& 2 &{\small\color{red}\quad\quad \text{Substitute 8 for }x.} \cr |3|\overset{?}{<}& 2 &{\small\color{red}\quad\quad \text{Subtract.}} \cr 3\not<& 2 &{\small\color{red}\quad\quad \text{Solution does not check. ✗}} } $ |
|