Quadratic Inequalities

A quadratic inequality like $x^2 - x - 6 < 0$ asks where the parabola sits below (or above) the x-axis. The zeros split the number line into intervals to test.

Find the roots, mark them on a number line, and check a point in each interval. Where the inequality holds, that interval is part of the solution.

Find the zeros first

Solve the related equation $ax^2 + bx + c = 0$ by factoring or the formula. Those roots are where the parabola crosses the x-axis.

The roots divide the number line into intervals. The parabola keeps a single sign — all positive or all negative — within each interval.

The upward parabola below shows the idea: between its two zeros it dips below the x-axis (negative), and outside them it rises above (positive) — exactly the sign information an inequality asks for.

Test each interval

Pick a test value inside each interval and plug it into the quadratic. A negative result means the parabola is below zero there; positive means above.

Keep the intervals that match your inequality. Use brackets for $\leq$ / $\geq$ (roots included) and parentheses for strict inequalities.

Worked examples

Example 1: a 'less than' inequality

Solve $x^2 - x - 6 < 0$ .

Factor to find zeros

(x - 3)(x + 2) = 0 \Rightarrow x = 3, -2

Test between the roots (x = 0)

0 - 0 - 6 = -6 < 0 ;\checkmark

Solution is the middle interval

-2 < x < 3

Answer: $-2 < x < 3$

Example 2: choosing bracket type

For $x^2 - x - 6 \leq 0$ , are the roots included?

The inequality allows equality

\leq

So the roots are part of the solution

[-2, 3]

Answer: Yes: $[-2, 3]$

Example 3: a 'greater than' inequality

Solve $x^2 - x - 6 > 0$ .

Factor to find zeros

(x - 3)(x + 2) = 0 \Rightarrow x = 3, -2

Test outside the roots (x = 4)

16 - 4 - 6 = 6 > 0

Both outer intervals work

x < -2 \text{ or } x > 3

Answer: $x < -2$ or $x > 3$

Try one yourself

Common questions

Why find the zeros first?

The zeros are the only places the parabola can switch between positive and negative. They mark the boundaries of the intervals you test.

How do I know which interval to keep?

Test one point per interval in the quadratic. Keep intervals whose sign matches the inequality (negative for $< 0$ , positive for $> 0$ ).

When are the roots included?

For $\leq$ or $\geq$ the boundary points satisfy the inequality, so they are included (use brackets). For strict $<$ or $>$ they are not.

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Quadratic Inequalities

Find the zeros first

Test each interval

Worked examples

Example 1: a 'less than' inequality

Example 2: choosing bracket type

Example 3: a 'greater than' inequality

Try one yourself

Common questions

Keep learning

Graphing Quadratic Functions

Solving Quadratic Equations by Factoring

Two-Step Inequalities