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AlgebraFoundation and Higher (some Higher only)

Inequalities GCSE questions with answers

8 inequalities questions to try, each with the answer and the full working one tap away. Every question here was made and worked out by the Show Working maths engine, not typed in by hand.

Inequality questions ask you to solve inequalities like 7 ≤ 4x + 9 < 13 and to list the whole numbers that fit one. On Higher you also solve quadratic inequalities, such as x² < 25.

Quiz yourself: 8 questions

Worked example 1: list the integers in an inequality

n is an integer and −7 < n < −3. Write down all the possible values of n.

  1. −7 is not included (the sign is <). −3 is not included (the sign is <).
  2. The integers that fit are −6, −5, −4.

Answer −6, −5, −4

01 / Quiz yourselfTry it, then tap to check

Inequalities questions to try

Work each one out on paper first. Then open the answer and compare your working with ours, step by step.

01 / List the integers in an inequalityBoth tiers

n is an integer and 1 < n ≤ 5. Write down all the possible values of n.

Show the answer and working
  1. 1 is not included (the sign is <). 5 is included (the sign is ≤).
  2. The integers that fit are 2, 3, 4, 5.

Answer 2, 3, 4, 5

02 / List the integers in an inequalityBoth tiers

x is an integer and 0 ≤ x < 6. Write down all the possible values of x.

Show the answer and working
  1. 0 is included (the sign is ≤). 6 is not included (the sign is <).
  2. The integers that fit are 0, 1, 2, 3, 4, 5.

Answer 0, 1, 2, 3, 4, 5

03 / List the integers in an inequalityBoth tiers

The number line shows an inequality for x. x is an integer. Write down all the possible values of x.

123456789101112
Show the answer and working
  1. A filled circle means the number is included; an open circle means it is not.
  2. So the inequality is 3 ≤ x < 10.
  3. The integers that fit are 3, 4, 5, 6, 7, 8, 9.

Answer 3, 4, 5, 6, 7, 8, 9

04 / Solve linear inequalitiesBoth tiers

Solve −10 ≤ 4x − 8 ≤ 4.

Show the answer and working
  1. Do the same to all three parts. Add 8: −2 ≤ 4x ≤ 12.
  2. Divide all three parts by 4: −12 ≤ x ≤ 3.

Answer −12 ≤ x ≤ 3

05 / Solve linear inequalitiesBoth tiers

Solve 9x + 4 < −4.

Show the answer and working
  1. Subtract 4 from both sides: 9x < −8.
  2. Divide both sides by 9 (a positive number, so the sign stays the same): x < −89.

Answer x < −89

06 / Solve linear inequalitiesBoth tiers

Solve 3(x + 6) ≤ 6.

Show the answer and working
  1. Expand the bracket: 3x + 18 ≤ 6.
  2. Subtract 18 from both sides: 3x ≤ −12.
  3. Divide both sides by 3: x ≤ −4.

Answer x ≤ −4

07 / Quadratic inequalitiesHigher only

Solve x2 ≤ 1.

Show the answer and working
  1. Get 0 on one side: x2 − 1 ≤ 0. Factorise: (x − 1)(x + 1) ≤ 0.
  2. The graph of y = x2 − 1 is a U shape crossing the x-axis at x = −1 and x = 1.
  3. It is on or below the x-axis between those two values, so −1 ≤ x ≤ 1.

Answer −1 ≤ x ≤ 1

08 / Quadratic inequalitiesHigher only

Solve x2 − x − 2 < 0.

Show the answer and working
  1. Factorise: (x + 1)(x − 2) < 0.
  2. The graph of y = x2 − x − 2 is a U shape crossing the x-axis at x = −1 and x = 2.
  3. It is below the x-axis between those two values, so −1 < x < 2.

Answer −1 < x < 2

02 / What's covered3 kinds of question

Inequalities at GCSE and where marks go

  • List the integers in an inequalityBoth tiers
  • Solve linear inequalitiesBoth tiers
  • Quadratic inequalitiesHigher only

Worked example 2: quadratic inequalities

Solve x2 − x − 56 ≥ 0.

  1. Factorise: (x + 7)(x − 8) ≥ 0.
  2. The graph of y = x2 − x − 56 is a U shape crossing the x-axis at x = −7 and x = 8.
  3. It is on or above the x-axis outside those two values, so x ≤ −7 or x ≥ 8.

Answer x ≤ −7 or x ≥ 8

A common mistake

Not turning the sign round when you divide by a negative number. If −2x > 6, dividing both sides by −2 gives x < −3, not x > −3.

03 / Practise itiPhone and iPad

More inequalities questions in Show Working

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More algebra topics

SubstitutionSimplifying expressionsPowers and indicesExpanding bracketsFactorisingSolving equationsSequencesSimultaneous equationsQuadratic equationsRearranging formulaeStraight-line graphsQuadratic and other graphsReal-life graphsFunctionsIterationAll GCSE maths topics
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