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GCSE Maths Inequalities Notes

An inequality compares two expressions that are not necessarily equal. You solve them like equations, but must flip the sign when multiplying or dividing by a negative.

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Key concepts

Inequality symbols
< less than, > greater than, ≤ less than or equal to, ≥ greater than or equal to.
Solving inequalities
Use the same steps as equations — the answer is a range of values, not a single number.
Flipping the sign
Multiplying or dividing both sides by a negative reverses the inequality symbol.
Number lines
An open circle means the value is not included (< or >); a filled circle means it is (≤ or ≥).

Key formulas

Symbols

< ≤ ≥ >

Worked example

Solve 3x − 4 ≤ 11.

  1. 1Add 4 to both sides: 3x ≤ 15.
  2. 2Divide both sides by 3: x ≤ 5.
  3. 3Show on a number line with a filled circle at 5 and an arrow to the left.

Answer: x ≤ 5

Common mistakes

✗ Forgetting to flip the sign when dividing by a negative.

✓ −2x < 6 becomes x > −3, not x < −3.

✗ Using the wrong type of circle on a number line.

✓ Open for < and >; filled for ≤ and ≥.

✗ Writing a single number instead of a range.

✓ The answer to an inequality is a set of values, e.g. x ≤ 5.

Quick check

Try these, then reveal the answers.

Solve x + 2 > 6.Reveal answer

x > 4

Solve 2x ≤ 10.Reveal answer

x ≤ 5

Solve −x < 3.Reveal answer

x > −3

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Exam tips

  • Treat solving like an equation, but watch for negatives.
  • List integer solutions carefully when a question asks for whole numbers in a range.
  • Double-check the direction of the inequality in your final answer.

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Related topics

AlgebraLinear Equations

Key terms explained

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What is area under a curve?What is chain rule?What is completing the square?What is definite integral?What is differentiation?What is discriminant?Full glossary

Frequently asked questions

When do I flip the inequality sign?+

Only when you multiply or divide both sides by a negative number.

What does an open circle mean on a number line?+

The value is not included — it goes with the strict symbols < and >.

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