compound inequality

A compound inequality is two simple inequalities tied together with the word and or the word or. Think of it as setting two filters at once: with and, a value must pass both filters; with or, passing either one is enough.

An and inequality describes an overlap — the values caught between two bounds, like 2 < x ≤ 7, which can be written as a single double inequality. Its solution is the intersection of the two pieces. An or inequality describes a union — values that satisfy at least one part, such as x < −1 or x > 4, often leaving a gap in the middle.

On a number line, and shades the region shared by both, while or shades everything either part covers. Translating the right English word into intersection or union is the heart of getting these correct.