Compound inequalities

A compound inequality combines two simple inequalities.

• AND (intersection): both must hold. Example: −2 < x ≤ 5 is shorthand for x > −2 AND x ≤ 5.
• OR (union): at least one must hold. Example: x < −1 OR x > 4.

Strategy

Solve each piece independently, then combine on a number line.

3x − 1 > 5 AND 3x − 1 ≤ 14 → x > 2 AND x ≤ 5 → 2 < x ≤ 5.