Inequalities with absolute values

Inequalities with absolute values can be rewritten as compound inequalities. For $|x| < a$ (where $a > 0$), the solution is $-a < x < a$. For $|x| > a$, the solution is $x < -a$ or $x > a$. If the absolute value is of an expression, like $|x-b| < a$, the solution is $b-a < x < b+a$.

Important properties

• For $|A| < B$ (with $B > 0$), the solution is $-B < A < B$.

• For $|A| > B$ (with $B > 0$), the solution is $A < -B$ or $A > B$.

• If $B < 0$, $|A| < B$ has no solution, and $|A| > B$ is always true.

• Absolute value inequalities can be visualized on the number line.