Quadratic inequalities

A quadratic changes sign at a simple root. A repeated root touches zero without changing sign. Make a sign diagram or test one point in each interval; do not assume that solving an inequality means taking the interval between the roots.

Use closed endpoints for ≤ or ≥ when equality is permitted. If you multiply or divide by a negative number, reverse the inequality. Multiplying by an expression of unknown sign needs separate cases, which is why a sign diagram is often safer for rational inequalities.

Worked example

Solve x² − x − 6 > 0.

1. Factor as (x − 3)(x + 2).
2. The roots are −2 and 3. The positive leading coefficient means the curve is above the axis outside these roots.
3. Strict inequality excludes both endpoints.

Answer: x < −2 or x > 3

Revise first: Solving quadratic equations.