Free GCSE Maths lesson: Algebra
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Lesson 31 · GCSE / Key Stage 4 · Maths · Algebra

Inequalities

Learn how to compare values, draw solution sets and solve inequalities without losing the direction of the sign.

Qualification: GCSE Key Stage 4 Subject: Maths Strand: Algebra

GCSE specification fit

A core algebra skill for comparing possible values.

Inequalities describe ranges rather than single answers. GCSE questions may ask you to solve them, draw them on a number line, or list the integer values that work.

QualificationGCSE Mathematics
Key stageKey Stage 4
StrandAlgebra
Tier guidanceFoundation and Higher

What you will learn

  • How to read <, >, ≤ and ≥.
  • How to draw open and closed endpoints on number lines.
  • How to solve one-step and two-step inequalities.
  • When to reverse the inequality sign.
  • How to list integer solutions.
  • How to interpret simple double inequalities and real-life constraints.

Why this matters

Inequalities appear when an answer can be more than one value: a cost must be under a budget, a measurement must be within a range, or a variable can take several integer values.

The method is close to solving equations, but the final answer is a range. That range must be written and drawn accurately.

Prior knowledge

You should already be comfortable with:

  • ordering positive and negative numbers,
  • reading inequality symbols,
  • using inverse operations,
  • solving linear equations,
  • multiplying and dividing by negative numbers.

Clear explanation

Read the symbol from the variable

Read an inequality from left to right. The small end points to the smaller value.

x > 4 means x is greater than 4 x < 4 means x is less than 4 x ≥ 4 means x is greater than or equal to 4 x ≤ 4 means x is less than or equal to 4

Open and closed circles

Use an open circle for < or > because the endpoint is not included. Use a closed circle for ≤ or ≥ because the endpoint is included.

x > 2: open circle at 2, shade to the right x ≥ 2: closed circle at 2, shade to the right x < 2: open circle at 2, shade to the left x ≤ 2: closed circle at 2, shade to the left

Solve like an equation

Use inverse operations on both sides. Most steps work just like equation solving.

3x + 5 < 20 3x < 15 x < 5

Reverse the sign when dividing by a negative

Multiplying or dividing both sides by a negative reverses the order of the numbers, so the inequality sign must reverse too.

−2x < 10 x > −5 The sign reverses because both sides were divided by −2

Integer solutions

If a question asks for integer solutions, list whole-number values only. Check the endpoint carefully.

−2 ≤ x < 3 Integer solutions: −2, −1, 0, 1, 2

Use the context to choose sensible values

In worded questions, the algebraic range is not always the final answer. A context may require whole numbers, positive values or a maximum allowed amount.

A lift can carry at most 400 kg If each box is 35 kg and the trolley is 50 kg: 35b + 50 ≤ 400 35b ≤ 350, so b ≤ 10 The maximum whole number of boxes is 10

Worked examples

Example 1: Draw a simple inequality

Represent x ≥ 6 on a number line.

Closed circle at 6 because 6 is included. Shade to the right because x is greater than or equal to 6.
Answer: closed circle at 6, arrow to the right.

Example 2: Solve a two-step inequality

Solve 4x − 3 ≤ 17.

4x ≤ 20 x ≤ 5
Answer: x ≤ 5.

Example 3: Reverse the sign

Solve −3y ≥ 12.

Divide both sides by −3. y ≤ −4
Answer: y ≤ −4.

Example 4: List integer solutions

List the integer solutions of −1 < n ≤ 4.

n must be bigger than −1, but 4 is allowed.
Answer: 0, 1, 2, 3, 4.

Quick checks

Choose an answer, then check your thinking.

1. How should you read x ≤ 7 on a number line?

2. Solve 2x + 1 > 9.

3. Solve −5x < 20.

Practice questions

Question 1

Write in words: x > 12.

Reveal answer and marking guidance

Answer: x is greater than 12.

Marking: Do not include 12, because there is no equals line under the symbol.

Question 2

Describe the number line for x ≤ −3.

Reveal answer and marking guidance

Answer: closed circle at −3, shaded to the left.

Marking: Use a closed circle because −3 is included, and shade left for smaller values.

Question 3

Solve x + 9 ≥ 15.

Reveal answer and marking guidance

Answer: x ≥ 6.

Marking: Subtract 9 from both sides and keep the inequality direction unchanged.

Question 4

Solve 3a − 4 < 11.

Reveal answer and marking guidance

Answer: a < 5.

Marking: Add 4 to get 3a < 15, then divide by 3.

Question 5

Solve −2p ≥ 14.

Reveal answer and marking guidance

Answer: p ≤ −7.

Marking: Divide by −2 and reverse the inequality sign.

Question 6

List the integer solutions of −2 < n ≤ 3.

Reveal answer and marking guidance

Answer: −1, 0, 1, 2, 3.

Marking: Exclude −2 because the left symbol is strict, but include 3 because the right symbol has equals.

Question 7

Solve 5 − 2x < 13.

Reveal answer and marking guidance

Answer: x > −4.

Marking: Subtract 5 to get −2x < 8, then divide by −2 and reverse the sign.

Question 8

A theatre trip has a coach budget of £240. Each ticket costs £18 and there is a fixed booking fee of £24. Write and solve an inequality for the maximum number of tickets.

Reveal answer and marking guidance

Answer: 18t + 24 ≤ 240, so t ≤ 12. The maximum is 12 tickets.

Marking: Subtract 24 to get 18t ≤ 216, divide by 18, and interpret the answer as a whole-number maximum.

Question 9

Solve −1 ≤ 2x + 3 < 9, then list the integer solutions.

Reveal answer and marking guidance

Answer: −2 ≤ x < 3, with integer solutions −2, −1, 0, 1, 2.

Marking: Subtract 3 from all three parts to get −4 ≤ 2x < 6, then divide all parts by 2. Include −2 because the left endpoint is closed, and exclude 3 because the right endpoint is strict.

Question 10

A lift can carry at most 630 kg. One adult has mass 78 kg and each box has mass 36 kg. Write and solve an inequality for the maximum number of boxes.

Reveal answer and marking guidance

Answer: 78 + 36b ≤ 630, so b ≤ 15.333... The maximum is 15 boxes.

Marking: Use ≤ because the lift can carry exactly 630 kg. Subtract 78 to get 36b ≤ 552, divide by 36, then round down because only whole boxes are possible.

Answers and marking guidance

The exact practice answers are hidden under each question so you can try first. For inequalities, marks usually come from reading the symbol correctly, showing balanced inverse operations, reversing the sign when multiplying or dividing by a negative number, using open or closed endpoints correctly, handling double inequalities across all parts of the statement, and interpreting the final range in the context when only whole-number values make sense.

Common mistakes

  • Including a strict endpoint: x < 5 does not include 5.
  • Using the wrong circle: open for < or >, closed for ≤ or ≥.
  • Forgetting to reverse the sign: reverse it when multiplying or dividing by a negative number.
  • Listing non-integers: if the question asks for integer solutions, list whole-number values only.
  • Ignoring the context: a ticket, person or box count must usually be a whole number inside the inequality range.
  • Reading the symbol backwards: say the inequality aloud from left to right before answering.

Extension challenge

Solve −3(2x − 1) ≤ 15.

Reveal answer

Answer: x ≥ −2.

Expand to −6x + 3 ≤ 15, then −6x ≤ 12. Dividing by −6 reverses the sign, so x ≥ −2.

Exam-board guidance

Inequalities are assessed by every GCSE Maths board. They can appear as direct algebra, number-line questions, integer-solution questions or constraints inside worded problems.

AQA GCSE Maths

State the final range clearly, use open circles for strict inequalities and closed circles when the endpoint is included, then check any requested integer list.

OCR GCSE Maths

Keep each balancing step visible, and explicitly reverse the inequality sign when multiplying or dividing by a negative number.

Pearson Edexcel GCSE Maths

If asked for integer solutions, list only whole-number values inside the final range and check whether each endpoint is included.

Eduqas GCSE Maths

Read the symbol aloud from the variable, then check whether the endpoint is included before drawing or listing the solution set.

WJEC Wales

Inequalities can describe limits such as maximum cost, minimum age or capacity, so interpret the final range in the real-life context before choosing values.

CCEA GCSE Maths

Show the inequality working clearly and check whether the unit question wants a range, a number line or integer answers.

Next lesson

Next, practise using formulae and rearranging them so the letter you need is on its own.

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