Solving inequalities and showing them on a number line
Solve linear inequalities, including double inequalities, and remember to reverse the sign when dividing by a negative, with worked examples.
GCSEHigh SchoolCurriculum fit: GCSE Maths Algebra (Foundation and Higher). US: Grade 7-8 and Algebra I inequalities (Reasoning with Equations and Inequalities).
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What is it?
An inequality compares two values that are not necessarily equal. The symbols are < (less than), > (greater than), ≤ (less than or equal to) and ≥ (greater than or equal to).
Linear inequalities are solved almost exactly like linear equations. The one difference is what happens when you multiply or divide by a negative number.
Key formulas and ideas
- Solve using inverse operations, as with equations.
- If you multiply or divide both sides by a negative number, reverse the inequality sign.
- On a number line, use an open circle for < and >, and a filled circle for ≤ and ≥.
Worked examples
Example 1: A basic inequality
Question: Solve 3x + 2 < 11.
- Subtract 2: 3x < 9.
- Divide by 3: x < 3.
Answer: x < 3
Example 2: Dividing by a negative
Question: Solve −2x > 8.
- Divide both sides by −2.
- Reverse the sign because you divided by a negative: x < −4.
Answer: x < −4
Example 3: Double inequality with integers
Question: List the integers satisfying −3 < 2x + 1 ≤ 9.
- Subtract 1 from all parts: −4 < 2x ≤ 8.
- Divide all parts by 2: −2 < x ≤ 4.
- Integers: −1, 0, 1, 2, 3, 4.
Answer: −1, 0, 1, 2, 3, 4
Common mistakes
- Forgetting to reverse the sign when dividing by a negative number.
- Using the wrong circle on a number line (open or filled).
- Including −2 in the answer to −2 < x, which excludes it.
- Treating the answer as a single value instead of a range.
Practice questions
Frequently asked questions
Why does the sign reverse for negatives?
For example 2 < 5, but multiplying both by −1 gives −2 > −5. Multiplying by a negative flips the order.
How do I check an inequality answer?
Choose a value in your solution range and substitute it into the original inequality. It should be true.
Related guides
Solving linear equations step by step (with examples)
Learn the balance method for solving linear equations, including brackets and x on both sides, with worked examples for GCSE and Algebra 1.
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KS3GCSEMiddle SchoolHigh SchoolSources and review
- DfE: GCSE mathematics subject content and assessment objectives
- Common Core State Standards: Mathematics (high school)
Reviewed by StudyMorrow editorial review on 5 Nov 2026.