Lesson 6: Solving Absolute Value Inequalities

Property

The absolute value of $x$ is defined by

Absolute value bars act like grouping devices in the order of operations: you should complete any operations that appear inside absolute value bars before you compute the absolute value.

Examples

• To evaluate $|-9|$, since $-9 < 0$, we use the second case of the definition: $|-9| = -(-9) = 9$.

• To evaluate the expression $10 - |5 - 8|$, first compute the operation inside the absolute value bars: $5 - 8 = -3$. Then take the absolute value: $|-3| = 3$. Finally, subtract: $10 - 3 = 7$.

Property

For any algebraic expression, $u$, and any positive real number, $a$:

• If $|u| < a$, then $-a < u < a$ (compound "and" inequality)
• If $|u| > a$, then $u < -a$ or $u > a$ (compound "or" inequality)

To solve an absolute value inequality, first isolate the absolute value expression, then apply the appropriate property based on the inequality symbol.

Examples

Property

When solving absolute value inequalities of the form $|x| < a$ (where $a > 0$), we get a compound "and" inequality: $-a < x < a$.
This means we need values of $x$ that satisfy both $x > -a$ AND $x < a$ simultaneously. The solution is the intersection of these two conditions.

Examples