Solving linear inequalities revision notes

Edexcel International GCSE Maths (4MA1), chapter 8: Inequalities, specification point 2.8c-f. Free lesson.

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Solve a linear inequality by rearranging it like an equation, but reverse the inequality sign when multiplying or dividing by a negative number. Leave the solution as an inequality, which describes all the values that work.

How to rearrange an inequality

You can add, subtract, multiply or divide both sides just as you would with an equation. Adding or subtracting the same amount keeps the order of the two sides unchanged, so these operations never flip the inequality sign. The final answer must remain an inequality, not just a single number.

When the inequality sign flips

Multiplying or dividing both sides by a negative number reverses the order, so you must reverse the inequality sign too. Multiplying by a negative reflects numbers across zero, changing which number is greater. A positive divisor leaves the sign as it is. Adding or subtracting a negative number does not flip the sign.

Solve an inequality with a positive divisor

Solve 3x + 4 < 19 by undoing the addition first, then dividing by 3. Since 3 is positive, the sign stays the same.

Solve an inequality with a negative divisor

Solve 5 − 2x ≥ 11 by subtracting 5 from both sides. Dividing by −2 reverses the sign, so ≥ becomes ≤. The answer means any value from −3 downwards works.

Check the answer and avoid the common mistake

If you divide −2x ≥ 6 by −2 but keep the sign, you get x ≥ −3, which is wrong. Testing x = 0 gives 0 ≥ 6, which is false. The correct answer is x ≤ −3; for example, x = −4 works. You can also check the last step of 7 − 3x < 1: dividing by −3 flips the sign, giving x > 2. Always reverse the sign when dividing by a negative. Testing a value can help check whether your answer makes sense.