Systems of inequalities in one variable

A system asks for values of x that satisfy two or more inequalities together. Solve each one, then take the values common to all, which is the intersection of the solution sets.

Worked example: 2x − 3 < x + 4 and 3 − 2x ≤ −1. The first gives x < 7. The second gives −2x ≤ −4, so x ≥ 2. Together: 2 ≤ x < 7, that is, [2, 7).

On the number line, shade each solution set on its own line and keep only the stretch shaded on both. The ends keep their own circle types, filled at 2 and hollow at 7.

A system can have no solution. x + 2 > 7 gives x > 5 and 2x < 4 gives x < 2; no number is both above 5 and below 2, so the solution set is empty, ∅.

A system can also collapse to one point: x ≥ 3 and x ≤ 3 together give only x = 3.

A double inequality is a system in short form: 1 ≤ x < 6 is the same as x ≥ 1 and x < 6.