Inequalities - AQA GCSE Mathematics Revision

Topic overview

What you need to know

Inequalities is a key part of GCSE Mathematics. Build fluency with the method, then apply it to unfamiliar and multi-step problems. A mathematical statement showing that one expression is greater than, less than, or not equal to another. Uses <, >, ≤, ≥. Solve like an equation but reverse the inequality sign when multiplying or dividing by a negative number. Show solutions on a number line: open circle (○) for < or >, filled circle (●) for ≤ or ≥.

Exam tip

Write every stage of your method. Method marks can still be earned when the final answer is wrong.

Revision notes

Core ideas

Secure these ideas first. Say each definition in your own words, then connect it to the topic overview.

Inequality: A mathematical statement showing that one expression is greater than, less than, or not equal to another. Uses <, >, ≤, ≥.
Solving Linear Inequalities: Solve like an equation but reverse the inequality sign when multiplying or dividing by a negative number.
Number Line Representation: Show solutions on a number line: open circle (○) for < or >, filled circle (●) for ≤ or ≥.
Double Inequality: An inequality with a variable between two bounds, e.g. −3 < x ≤ 5. Solve the middle part.

Revision notes

Apply it in the exam

The exam will rarely ask for an isolated definition. Practise selecting the right idea and using it as part of a complete explanation or method.

Graphical Inequalities: Shade the region satisfying the inequality on a coordinate grid. Use solid lines for ≤/≥ and dashed for </> .
Set Notation: Writing solution sets using braces. E.g. {x : x > 3} means the set of all x values greater than 3.
Quadratic Inequalities: Solve by finding roots, sketching the graph, and reading off where the curve is above or below zero.
Integer Solutions: When asked for integer values satisfying an inequality, list only the whole numbers in the range.

Core knowledge

Key facts for Inequalities

8 essentials

Inequality

A mathematical statement showing that one expression is greater than, less than, or not equal to another. Uses <, >, ≤, ≥.

Solving Linear Inequalities

Solve like an equation but reverse the inequality sign when multiplying or dividing by a negative number.

Number Line Representation

Show solutions on a number line: open circle (○) for < or >, filled circle (●) for ≤ or ≥.

Double Inequality

An inequality with a variable between two bounds, e.g. −3 < x ≤ 5. Solve the middle part.

Graphical Inequalities

Shade the region satisfying the inequality on a coordinate grid. Use solid lines for ≤/≥ and dashed for </> .

Set Notation

Writing solution sets using braces. E.g. {x : x > 3} means the set of all x values greater than 3.

Quadratic Inequalities

Solve by finding roots, sketching the graph, and reading off where the curve is above or below zero.

Integer Solutions

When asked for integer values satisfying an inequality, list only the whole numbers in the range.

Active recall

Close the notes and answer these

1.Without looking, explain inequality and give one example or consequence.
2.Without looking, explain solving linear inequalities and give one example or consequence.
3.Without looking, explain number line representation and give one example or consequence.
4.Without looking, explain double inequality and give one example or consequence.

Content reviewed 23 July 2026 against the current linked specification.

Finished learning?

Mark the notes complete, then test what you can recall.

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