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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 $\left|x\right|<4$ means that the distance between $x$ and $0$ is less than $4$ .

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

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 $\left|a\right|<b$ , then $a<b$ AND $a>-b$ .

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 $\left\{x|x<-4\text{ or }x>4\right\}$ .

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

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

In other words, for any real numbers $a$ and $b$ , if $\left|a\right|>b$ , then $a>b$ AND $a<-b$ .

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: