# 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 | x | < 4 means that the distance between x and 0 is less than 4.

So, x > –4 AND x < 4. The solution set is .

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 | a | < b, then a < b AND a > – b .

Example 1:

Solve and graph.

|x – 7| < 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

–3 + 7 < x – 7 + 7 < 3 + 7

4 < x <10

The graph looks like this:

## Absolute Value Inequalities (>):

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

So, x < –4 OR x > 4. The solution set is .

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 | a | > b, then a > b OR a < – b .

Example 2:

Solve and graph.

Split into two inequalities.

Subtract 2 from each sides of each inequality.

The graph looks like this: