LearnovaAll Maths notes
AQA MathsAlgebra

Linear inequalities

Solve and represent inequalities on number lines.

AQAGCSE MathsAlgebraFoundation and Higher
Visual model

Inequalities describe a range of values

x>3x>3open circle: 3 is not included
Gold-standard guide
20 mins

What you will learn

Solve and represent inequalities on number lines.
Use a clear step-by-step method for linear inequalities.
Check your answer and avoid the most common exam mistake.
Useful before you start
Core number skillsEarlier algebra skillsShowing clear working
Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

Method

Treat an inequality like an equation but keep the inequality sign

Step 1

Subtract 3 from both sides

−2x ≤ 11 − 3, so −2x ≤ 8

Step 2

Divide both sides by −2 — the inequality FLIPS because we divide by a negative

x ≥ −4

Step 3

Draw on a number line

A solid circle (filled dot) at −4 with an arrow pointing to the right, since ≥ includes −4

Watch out

Watch out

Students forget to flip the inequality when dividing by a negative number

f
Inequality rule

When multiplying or dividing by a negative, reverse the inequality sign.

f
Number line

Open circle means not included; closed circle means included.

Worked example

Solve 3 − 2x ≤ 11 and represent your solution on a number line

1

Subtract 3 from both sides: −2x ≤ 11 − 3, so −2x ≤ 8.

2

Divide both sides by −2 — the inequality FLIPS because we divide by a negative: x ≥ −4. The ≤ sign becomes ≥.This flip happens because dividing by a negative number reverses the order of the number line.

3

Draw on a number line: A solid circle (filled dot) at −4 with an arrow pointing to the right, since ≥ includes −4.

Final answer

x ≥ −4

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

Solve 3 − 2x ≤ 11 and represent your solution on a number line

3 marks4 minslinear-inequalities-worked
Show solution
Worked solution
  1. 1.Subtract 3 from both sides: −2x ≤ 11 − 3, so −2x ≤ 8.
  2. 2.Divide both sides by −2 — the inequality FLIPS because we divide by a negative: x ≥ −4. The ≤ sign becomes ≥.This flip happens because dividing by a negative number reverses the order of the number line.
  3. 3.Draw on a number line: A solid circle (filled dot) at −4 with an arrow pointing to the right, since ≥ includes −4.
Final answer

x ≥ −4

Mark points
  • M1: subtract 3 from both sides
  • M1: divide both sides by −2 — the inequality flips because we divide by a negative
  • M1: draw on a number line
  • A1: x ≥ −4
Watch out

Students forget to flip the inequality when dividing by a negative number.This is the single most common error in linear inequality questions.One way to avoid it: instead of dividing by −2, add 2x to both sides and keep the coefficient positive throughout.For example: 3 − 2x ≤ 11 → 3 ≤ 11 + 2x → −8 ≤ 2x → x ≥ −4.

Diagnosticrecall

Solve 5x − 3 > 17

1 mark2 minslinear-inequalities-q1
Show solution
Worked solution
  1. 1.Spot the skill: Treat an inequality like an equation but keep the inequality sign.
  2. 2.Use the subtract 3 from both sides stage first, then divide both sides by −2 — the inequality flips because we divide by a negative.
  3. 3.Keep the final answer visible: x > 4.
Final answer

x > 4

Mark points
  • M1: use the correct treat an inequality like an equation but keep the inequality sign.critical rule: if you multiply or divide both sides by a negative number, the inequality sign flips direction.a handy memory aid: multiplying by a negative 'flips' the sign because the number line mirrors.
  • A1: x > 4
Watch out

Students forget to flip the inequality when dividing by a negative number.This is the single most common error in linear inequality questions.One way to avoid it: instead of dividing by −2, add 2x to both sides and keep the coefficient positive throughout.For example: 3 − 2x ≤ 11 → 3 ≤ 11 + 2x → −8 ≤ 2x → x ≥ −4.

Easyprocedure

Solve −3x < 9

2 marks3 minslinear-inequalities-q2
Show solution
Worked solution
  1. 1.Spot the skill: Treat an inequality like an equation but keep the inequality sign.
  2. 2.Use the divide both sides by −2 — the inequality flips because we divide by a negative stage first, then draw on a number line.
  3. 3.Keep the final answer visible: x > −3.
Final answer

x > −3

Mark points
  • M1: use the correct treat an inequality like an equation but keep the inequality sign.critical rule: if you multiply or divide both sides by a negative number, the inequality sign flips direction.a handy memory aid: multiplying by a negative 'flips' the sign because the number line mirrors.
  • A1: x > −3
Watch out

Students forget to flip the inequality when dividing by a negative number.This is the single most common error in linear inequality questions.One way to avoid it: instead of dividing by −2, add 2x to both sides and keep the coefficient positive throughout.For example: 3 − 2x ≤ 11 → 3 ≤ 11 + 2x → −8 ≤ 2x → x ≥ −4.

Mediumreasoning

Find all integer values of x satisfying −2 < x ≤ 3

3 marks4 minslinear-inequalities-q3
Show solution
Worked solution
  1. 1.Spot the skill: Treat an inequality like an equation but keep the inequality sign.
  2. 2.Use the draw on a number line stage first, then subtract 3 from both sides.
  3. 3.Keep the final answer visible: −1, 0, 1, 2, 3.
Final answer

−1, 0, 1, 2, 3

Mark points
  • M1: use the correct treat an inequality like an equation but keep the inequality sign.critical rule: if you multiply or divide both sides by a negative number, the inequality sign flips direction.a handy memory aid: multiplying by a negative 'flips' the sign because the number line mirrors.
  • A1: −1, 0, 1, 2, 3
Watch out

Students forget to flip the inequality when dividing by a negative number.This is the single most common error in linear inequality questions.One way to avoid it: instead of dividing by −2, add 2x to both sides and keep the coefficient positive throughout.For example: 3 − 2x ≤ 11 → 3 ≤ 11 + 2x → −8 ≤ 2x → x ≥ −4.

Hardproblem solving

Solve 4(x − 1) ≥ 2x + 6

3 marks5 minslinear-inequalities-q4
Show solution
Worked solution
  1. 1.Spot the skill: Treat an inequality like an equation but keep the inequality sign.
  2. 2.Use the subtract 3 from both sides stage first, then divide both sides by −2 — the inequality flips because we divide by a negative.
  3. 3.Keep the final answer visible: x ≥ 5.
Final answer

x ≥ 5

Mark points
  • M1: use the correct treat an inequality like an equation but keep the inequality sign.critical rule: if you multiply or divide both sides by a negative number, the inequality sign flips direction.a handy memory aid: multiplying by a negative 'flips' the sign because the number line mirrors.
  • A1: x ≥ 5
Watch out

Students forget to flip the inequality when dividing by a negative number.This is the single most common error in linear inequality questions.One way to avoid it: instead of dividing by −2, add 2x to both sides and keep the coefficient positive throughout.For example: 3 − 2x ≤ 11 → 3 ≤ 11 + 2x → −8 ≤ 2x → x ≥ −4.

Exam-stylemulti-step

Find all integer values of n satisfying both 2n − 1 > 3 and n² < 30

4 marks6 minslinear-inequalities-q5
Show solution
Worked solution
  1. 1.Spot the skill: Treat an inequality like an equation but keep the inequality sign.
  2. 2.Use the divide both sides by −2 — the inequality flips because we divide by a negative stage first, then draw on a number line.
  3. 3.Keep the final answer visible: 3, 4, 5.
Final answer

3, 4, 5

Mark points
  • M1: use the correct treat an inequality like an equation but keep the inequality sign.critical rule: if you multiply or divide both sides by a negative number, the inequality sign flips direction.a handy memory aid: multiplying by a negative 'flips' the sign because the number line mirrors.
  • A1: 3, 4, 5
Watch out

Students forget to flip the inequality when dividing by a negative number.This is the single most common error in linear inequality questions.One way to avoid it: instead of dividing by −2, add 2x to both sides and keep the coefficient positive throughout.For example: 3 − 2x ≤ 11 → 3 ≤ 11 + 2x → −8 ≤ 2x → x ≥ −4.

Grade 9 stretchproblem solving

Solve -3(2x - 1) > 15.

4 marks7 minsinequality-g9
Show solution
Worked solution
  1. 1.Expand the bracket.
  2. 2.Rearrange the inequality.
  3. 3.Reverse the sign when dividing by a negative number.
Final answer

x < -2

Mark points
  • M1: -6x + 3 > 15
  • M1: -6x > 12
  • A1: x < -2
Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Timed checkpoint
12 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Linear inequalities - 2 marksSolve 5x − 3 > 17Mark answer
Answer

x > 4

2Algebraic notation and substitution - 2 marksFind t² + 4t when t = −2Mark answer
Answer

−4

3Simplifying expressions - 2 marksSimplify 2xy + 3x²y − xy + x²yMark answer
Answer

3x²y + xy

4Expanding and factorising - 3 marksFactorise fully 6x² + 9xMark answer
Answer

3x(2x + 3)

Mastery check
  • I can explain the method for linear inequalities.
  • I can show clear working without skipping key steps.
  • I can avoid this mistake: Students forget to flip the inequality when dividing by a negative number.This is the single most common error in linear inequality questions.One way to avoid it: instead of dividing by −2, add 2x to both sides and keep the coefficient positive throughout.For example: 3 − 2x ≤ 11 → 3 ≤ 11 + 2x → −8 ≤ 2x → x ≥ −4.
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Related topics
Algebraic notation and substitutionSimplifying expressionsExpanding and factorising
Official exam-board sources

This guide follows the AQA GCSE Mathematics 8300 specification. Practice questions are original Learnova questions shaped around official content and exam skills.

Official specification Official past papers
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