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Absolute Value Inequalities

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

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

x – 7 < 3 AND – (x – 7) < 3

Add 7 to both sides of the first inequality.

x < 10 AND – (x – 7) < 3

Multiply both sides of the second inequality by –1. Remember to reverse the direction of the inequality.

x < 10 AND x – 7 > –3

Add 7 to both sides of the second inequality.

x < 10 AND x > 4

We can write this more simply as:

4 < x < 10

Its graph looks like this:

Some absolute value inequalities result in an "OR" inequality, rather than an "AND" inequality. The graphs of these will be two rays in opposite directions.

Split into two inequalities.

x + 2 4 OR – (x + 2) 4

Subtract 2 from both sides of the first inequality.

x 2 OR – (x + 2) 4

Multiply the second inequality by –1. Again, remember to reverse the direction.

x 2 OR x + 2 –4

Subtract 2 from both sides of the second inequality.

x 2 OR x – 6