Solving inequalities — Mathematics, 14–17 years
Inequalities describe a range of possible answers, not just one number. Learn how to solve them and show the result clearly on a number line.
An inequality keeps a range
An equation asks which value makes two expressions equal. An inequality asks which values make one expression larger or smaller, so its answer is usually a whole interval; on a number line, the possible values lie on one side of a boundary.
Why inequalities are useful
Many real questions have limits rather than exact targets: a bag may weigh at most 5 kg, or a ride may require someone taller than 140 cm. Inequalities grew from the need to reason about such safe ranges and restrictions without testing every possible value.
A worked example
Solve 3x + 2 ≤ 14. First subtract 2: 3x ≤ 12. Then divide by 3: x ≤ 4, so every number up to 4 works. Check x = 4: 3·4 + 2 = 14; check x = 5: 17 is not at most 14.
The reversed sign
When you multiply or divide an inequality by a negative number, the sign must reverse: −2x > 6 becomes x < −3. This feels odd because ordinary equality keeps its sign, but multiplying by a negative flips the order of numbers on the number line.
Limits in real decisions
Inequalities appear in budgets, safety rules and timetables. If you have €30 and each ticket costs €6, the condition 6n ≤ 30 tells you that you can buy at most five tickets; it does not claim that you must buy exactly five.
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