What you'll learn
Quadratic and other non-linear inequalities ask you to find the range of values that satisfy an inequality involving a quadratic (or similar) expression. In this guide you will learn how to solve quadratic inequalities by finding the critical values, how to use a sketch or sign analysis to choose the correct region, how to write your answer using inequality notation, and how to avoid common errors. This builds on solving quadratic equations.
Key terms and definitions
Quadratic inequality — an inequality involving a quadratic expression, e.g. x² − 5x + 6 > 0.
Critical values — the solutions of the corresponding quadratic equation (where the expression equals zero).
Parabola — the U-shaped (or ∩-shaped) curve of a quadratic.
Region — the set of x-values satisfying the inequality.
Inequality notation — symbols such as <, >, ≤, ≥ used to describe ranges.
Core concepts
Find the critical values
To solve a quadratic inequality, first solve the matching equation (set the expression equal to zero) to find the critical values. For x² − 5x + 6 > 0, solve x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → critical values x = 2 and x = 3. These split the number line into regions.
Sketch the parabola
A positive x² term gives a U-shaped parabola that crosses the x-axis at the critical values. The curve is below the x-axis between the roots and above outside them. Sketching the parabola makes it easy to see which region satisfies the inequality.
Choose the correct region
- For > 0 (or ≥ 0), you want where the curve is above the axis — usually outside the roots: x < 2 or x > 3.
- For < 0 (or ≤ 0), you want where the curve is below the axis — usually between the roots: 2 < x < 3.
The direction of the inequality and the shape of the parabola decide the region.
Writing the answer
Write your answer using inequality notation. A "between the roots" answer is a single inequality (2 < x < 3); an "outside the roots" answer needs two inequalities joined by "or" (x < 2 or x > 3). Use ≤ or ≥ if the original inequality includes "equal to".
A useful check
Pick a test value in a region and substitute it into the inequality. If it works, that region is part of the solution; if not, it isn't. This confirms you've chosen the right side.
Worked examples
Example 1: Less-than inequality
Solve x² − 5x + 6 < 0.
Critical values: x = 2 and x = 3. The U-shaped curve is below the axis between the roots, so 2 < x < 3.
Example 2: Greater-than inequality
Solve x² − x − 12 ≥ 0.
Factorise: (x − 4)(x + 3) = 0 → critical values −3 and 4. Above the axis is outside the roots: x ≤ −3 or x ≥ 4.
Example 3: Test value check
Confirm the solution to x² − 4 > 0.
Critical values ±2. Test x = 0: 0 − 4 = −4, not > 0, so the middle region fails. The solution is x < −2 or x > 2.
Common mistakes and how to avoid them
Treating it like an equation. An inequality gives a range of values, not just the critical values.
Choosing the wrong region. Sketch the parabola or use a test value to decide above/below the axis.
Writing "outside" answers as one inequality. Use two inequalities joined by "or".
Forgetting ≤ / ≥. Include "equal to" if the original inequality does.
Sign errors in factorising. Factorise carefully to get correct critical values.
Exam technique for Non-linear Inequalities
Solve the equation first to find critical values.
Sketch the parabola to see where it is above or below the axis.
Match the region to the inequality direction.
Use a test value to confirm.
Write the answer in correct notation, with "or" for outside-the-roots regions.
Quick revision summary
To solve a quadratic inequality, first find the critical values by solving the matching equation (setting the expression to zero and factorising). These split the number line into regions. Sketch the parabola: a positive x² term gives a U-shape that is below the axis between the roots and above outside them. For < 0, take the region between the roots (e.g. 2 < x < 3); for > 0, take the regions outside the roots (e.g. x < 2 or x > 3). Write "between" answers as a single inequality and "outside" answers as two inequalities joined by "or", including ≤ or ≥ if the original has "equal to". Confirm your choice with a test value. The main pitfalls are treating it like an equation, picking the wrong side, and writing outside-the-roots regions incorrectly. Find the critical values, sketch the curve, match the region to the inequality, test a value, and write the answer in proper notation.