Compound Inequalities

Join inequalities with and / or, then write the interval.

The idea

Definition (Compound inequality).

A compound inequality joins two inequalities with the word and or the word or. Its solution set is the intersection of the two pieces' solution sets when the word is and, and their union when the word is or.

We solve each piece in the usual way; the joining word then says how to combine the two solution sets.

And keeps the numbers satisfying both pieces — the intersection of the two sets. On a number line this is the stretch the two shadings share, a single piece running from some $a$ to some $b$, and it is empty when the sets never meet.

Or keeps the numbers satisfying at least one piece — the union, everything either shading covers. A union can be disconnected: $X < c$ or $X > d$, with $c < d$, leaves a gap between two rays.

A three-part statement $a < X < b$ abbreviates an and: $X$ is greater than $a$ and less than $b$ at once. We solve it whole, doing to both outer parts whatever we do to the middle.

We write answers in interval notation: a square bracket for an endpoint the set includes, a round parenthesis for one it excludes (or for an infinite end), and $\cup$ joining the pieces of a union.

Ways to work on it

• Walkthrough. Solve and / or inequalities and read off interval notation.
• Practice. Solve a three-part inequality and report a bound.
• Hardest. Solve both pieces of an and statement and combine them into one interval.

Not sure where to start? Take the ten-question placement test.