GCSE Maths / Edexcel

Solving linear inequalities

Solve linear inequalities, keep the inequality sign correct, and identify integer solutions where needed.

AlgebraFoundation and HigherGrades 4 to 6Skill

Curriculum path: GCSE Maths > Edexcel > Algebra > Linear inequalities

Pearson Edexcel GCSE Maths algebra A22: solve linear inequalities in one variable and represent solutions.

Revision notes

Theory, examples, and quick checks.

Keep the method short, then practise straight away. This note is written for GCSE Maths Edexcel students who need clear working and reliable method marks.

Theory

An inequality compares two expressions using signs such as <, >, <=, or >=.

< means less than, > means greater than, <= means less than or equal to, and >= means greater than or equal to.

Solve inequalities using the same balancing steps as equations: do the same operation to both sides.

The important extra rule is this: if you multiply or divide both sides by a negative number, reverse the inequality sign.

If the answer is x < 6, then x can be any number less than 6, not just one value.

When asked for integer solutions, list only whole numbers that fit the final inequality. Check whether the endpoint is included.

Key ruleReverse the inequality sign when multiplying or dividing by a negative.

Worked examples

One-step inequality

Solve x + 4 < 10.

  1. Subtract 4 from both sides.
  2. x < 6.

Answer: x < 6

Two-step inequality

Solve 3x - 2 >= 13.

  1. Add 2 to both sides: 3x >= 15.
  2. Divide by 3.

Answer: x >= 5

Negative coefficient

Solve -2x < 8.

  1. To get x on its own, divide both sides by -2.
  2. Because you are dividing by a negative, reverse the inequality sign.
  3. 8 / -2 = -4.

Answer: x > -4

Common mistakes

  • Treating < and > as if they mean equals.
  • Forgetting to reverse the sign after dividing by a negative.
  • Listing values that are not integers when the question asks for integer solutions.
  • Including the endpoint for < or > when it should only be included for <= or >=.

Quick exercise

Try these before moving to the exam-style questions.

  1. Solve x + 6 < 11.
  2. Solve 2x <= 14.
  3. Solve 5x - 3 > 12.
  4. Solve -4x >= 20.
  5. List the positive integer solutions to x < 4.
Exam-style questions

Practise the same skill at three levels.

These are original GCSE-style questions with mark schemes, common wrong answers, and AI marking guidance so feedback stays close to exam expectations.

Basic GCSE styleFoundationNon-calculator2 marks

Solve x + 5 <= 12.

inequalitiesinverse operationsfoundation algebra
Standard exam styleFoundation and HigherNon-calculator3 marks

Solve 4x - 3 > 9.

linear inequalitiestwo-step methodnumber line
ChallengeHigherEither4 marks

Solve -3x + 5 >= 14 and state the greatest integer solution.

negative coefficientinteger solutionshigher reasoning