# Absolute Value Inequalities

An
absolute value
**
inequality
**
is an
inequality
that has an absolute value sign with a variable inside.

## Absolute Value Inequalities ( $<$ ):

The inequality $\left|\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\right|<4$ means that the distance between $x$ and $0$ is less than $4$ .

So, $x>-4$ AND $x<4$ . The solution set is $\left\{x\text{\hspace{0.17em}}|\text{\hspace{0.17em}}-4<x<4\right\}$ .

When solving absolute value inequalities, there are two cases to consider.

Case $1$ : The expression inside the absolute value symbols is positive.

Case $2$ : The expression inside the absolute value symbols is negative.

The solution is the
**
intersection
**
of the solutions of these two cases.

In other words, for any real numbers $a$ and $b$ , if $\left|\text{\hspace{0.17em}}a\text{\hspace{0.17em}}\right|<b$ , then $a<b$ AND $a>-b$ .

**
Example 1
**
:

Solve and graph.

$\left|\text{\hspace{0.17em}}x-7\text{\hspace{0.17em}}\right|<3$

To solve this sort of inequality, we need to break it into a compound inequality .

$x-7<3$ and $x-7>-3$ .

$-3<x-7<3$

Add $7$ to each expression.

$\begin{array}{l}-3+7<x-7+7<3+7\\ 4<x<10\end{array}$

The graph looks like this:

## Absolute Value Inequalities ( $>$ ):

The inequality $\left|x\right|>4$ means that the distance between $x$ and $0$ is greater than $4$ .

So, $x<-4$ OR $x>4$ . The solution set is $\left\{x\text{\hspace{0.17em}}|\text{\hspace{0.17em}}x<-4\text{or}x4\right\}$ .

When solving absolute value inequalities, there are two cases to consider.

Case $1$ : The expression inside the absolute value symbols is positive.

Case $2$ : The expression inside the absolute value symbols is negative.In other words, for any real numbers $a$ and $b$ , if $\left|\text{\hspace{0.17em}}a\text{\hspace{0.17em}}\right|>b$ , then $a>b$ AND $a<-b$ .

**
Example 2
**
:

Solve and graph.

$\left|\text{\hspace{0.17em}}x+2\text{\hspace{0.17em}}\right|\ge 4$

Split into two inequalities.

$x+2\ge 4\text{OR}x+2\le -4$

Subtract $2$ from each sides of each inequality.

$x\ge 2\text{OR}x\le -6$

The graph looks like this: