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.
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.
- 1Add 4 to both sides: 3x ≤ 15.
- 2Divide both sides by 3: x ≤ 5.
- 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
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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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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