Absolute Value Inequalities

Two Types

Absolute value inequalities come in two forms, each with a different approach.

Type 1: Less Than

$|x| < 3$ means $x$ is less than 3 units from zero.

So $x$ is between $-3$ and $3$.

$|A| < B$ becomes $-B < A < B$

This is an “and” situation. Both conditions must be true.

Example

Type 2: Greater Than

$|x| > 3$ means $x$ is more than 3 units from zero.

So $x < -3$ or $x > 3$.

$|A| > B$ becomes $A < -B$ or $A > B$

This is an “or” situation. Either condition works.

Summary

Less than → trapped between two values

Greater than → escaped to the outer regions