An equation says two expressions have the same value. Solving finds the unknown that makes this true. Think of the equals sign as a balance: applying the same operation to both sides preserves equality. Undo additions and multiplications in a sensible order, or collect unknown terms first if they appear on both sides. An inequality describes a range of values instead of one value. The same balancing idea works, but multiplying or dividing both sides by a negative number reverses the inequality sign. Always check your equation solution by substituting it into the original question.
Learning objectives
- Solve a linear equation using balanced operations.
- Solve and represent a linear inequality.
Key ideas
Apply the same operation to both sides of an equation; check by substituting the solution.
Expand brackets, collect variable terms on one side, then isolate the unknown.
Strict inequalities exclude the boundary; inclusive inequalities include it.
Multiplying or dividing an inequality by a negative reverses the inequality sign.
On a number line: an open circle excludes the boundary; a filled circle includes it.
Worked examples
Example 1
Solve the equation by keeping both sides balanced.
Check both sides.
Answer
Example 2
Solve the inequality, reversing the sign when dividing by a negative.
Answer
Question styles to prepare for
Original practice guidance aligned to the specification, not a count of past-paper appearances or a prediction. The AO labels below describe the intended practice focus, not official AQA mark allocations.
AO1 · Methods
Solve an equation or inequality and represent the solution.
AO2 · Reasoning
Check a proposed solution and explain an invalid rearrangement.
AO3 · Problem solving
Form an equation from a word problem, solve it and interpret the answer.
When a task asks you to explain, include a mathematical reason. When solving a problem, show how the information becomes a calculation, equation or diagram, then interpret your result.
Avoid this mistake
Reverse the inequality sign only when multiplying or dividing by a negative, not when adding or subtracting.
Try it yourself
Attempt each question before opening its answer.
Check 1: Solve 4x - 7 = 13.
Answer: x = 5
Add 7 to both sides to get 4x = 20, then divide both sides by 4.
Check 2: Solve -2x < 8.
Answer: x > -4
Divide both sides by -2 and reverse the inequality sign.
Explain it in your own words
- Show a balance diagram for 2x + 3 = 11.
- Explain how to check a solution by substitution.
- Sketch x ≤ 3 and x < 3 and explain the different endpoint.
Keep practising
Download the compact topic guide (PDF) · Practise this topic in Blitz · All foundation notes
Specification source
AQA Algebra specification · Assessment objectives. Checked 16 October 2026. Topic references identify related content; these notes do not cover every specification statement.