KS3 Algebra Inequalities: Open Circles, Negative Numbers and Real-Life Limits

When children first learn algebra, they often solve equations such as x + 4 = 10. An equation asks for values that make two sides equal. An inequality asks which values make one side greater or smaller than the other.

This difference matters because an inequality usually has many solutions. The English mathematics national curriculum introduces the vocabulary of inequalities within Key Stage 3 algebra, while later GCSE work explicitly includes solving linear inequalities and representing solutions on number lines. Schools may introduce individual techniques at different points during Years 7–9, so this article is labelled KS3 rather than claiming that every school teaches it in one specific year.

1. Four symbols, four different meanings

• x < 5: x is less than five; five itself is not included.

• x ≤ 5: x is less than or equal to five; five is included.

• x > 5: x is greater than five.

• x ≥ 5: x is greater than or equal to five.

A useful way to read an inequality is to say it aloud before solving it. “At least five” means x ≥ 5. “More than five” means x > 5. These phrases are not interchangeable.

2. Why number lines use open and filled circles

Suppose x < 3. The solution includes numbers such as 2, 0 and −10, but not 3. On a number line, mark an open circle at 3 and shade to the left.

For x ≥ 3, mark a filled circle at 3 and shade to the right. The filled circle tells us that the endpoint itself is a solution.

Children sometimes memorise “the arrow points to the answer.” A stronger explanation is to test a number: does 2 satisfy x < 3? Yes. Does 4? No. The shading must include 2 and exclude 4.

3. Worked example: solve a simple inequality

Solve 3x + 5 ≤ 20.

Subtract five from both sides: 3x ≤ 15.

Divide both sides by three: x ≤ 5.

Check two values. If x = 5, the left side is 3(5) + 5 = 20, so the endpoint is included. If x = 6, the left side is 23, which is too large.

On a number line, draw a filled circle at 5 and shade left.

The algebra looks similar to solving an equation, but the answer is an entire set of values, not one number.

4. The surprising rule: reverse the sign for a negative multiplier or divisor

Solve 7 − 2x < 15.

Subtract seven: −2x < 8.

Now divide by −2. The inequality sign must reverse:

x > −4.

Why? Think of the true statement 2 < 5. Multiplying both sides by −1 gives −2 > −5, not −2 < −5. Multiplication by a negative reverses the order of the numbers on the number line.

Check the answer. With x = 0, the original left side is 7, and 7 < 15 is true. With x = −5, the left side is 17, and 17 < 15 is false. The solution x > −4 behaves exactly as expected.

Important: The sign reverses only when both sides are multiplied or divided by a negative number. Adding or subtracting a negative number does not automatically reverse it.

5. Compound inequalities: between two boundaries

Consider 3 ≤ 2x + 1 < 11.

This is a short way of saying two things at once: 2x + 1 is at least three, and it is less than eleven.

Subtract one from all three parts:

2 ≤ 2x < 10.

Divide every part by two:

1 ≤ x < 5.

On a number line, put a filled circle at 1, an open circle at 5, and shade the segment between them.

The endpoints matter. One is allowed; five is not. If a pupil writes “between one and five” without specifying inclusion, their answer is incomplete.

6. Real-life example: a budget creates an inequality

A parent has £60 to spend on an activity. There is an £18 registration fee and each ticket costs £7. What is the maximum number of tickets that can be purchased?

Let n be the number of tickets. The total cost must be no more than £60, so:

18 + 7n ≤ 60.

Subtract 18: 7n ≤ 42.

Divide by seven: n ≤ 6.

Because n is a nonnegative whole number, the maximum is six tickets. Six tickets cost 18 + 7(6) = £60. Seven tickets would cost £67 and exceed the budget.

Notice how the context changes the interpretation. Algebra allows many real values below six, but buying 4.5 tickets would not make sense here. We must combine the inequality with the real-world restriction that ticket counts are whole numbers.

7. Four common misconceptions

“The answer is always one number.” Unlike an equation such as x = 5, an inequality may describe infinitely many numbers. Test more than one candidate.

“The sign always stays the same.” Multiplying or dividing by a negative reverses it. Practise with simple number comparisons before tackling algebra.

“Open and filled circles mean the same thing.” They distinguish excluded and included endpoints. Ask whether the boundary itself works.

“Every mathematical solution makes sense in the story.” A number of tickets, students or buses must usually be a whole number. Interpret the answer in context.

8. Original practice questions with solutions

Question 1. Solve 5x − 4 ≥ 16. Answer: 5x ≥ 20, so x ≥ 4.

Question 2. Solve 9 − 3x ≤ 0. Answer: −3x ≤ −9. Divide by −3 and reverse the sign: x ≥ 3.

Question 3. Solve 2 < x + 5 ≤ 9. Answer: Subtract five throughout: −3 < x ≤ 4.

Question 4. A family pays £12 to enter an attraction and £4 per game. Their budget is £36. What is the maximum number of games? Answer: 12 + 4n ≤ 36, so 4n ≤ 24 and n ≤ 6. The maximum is six games.

Question 5. Describe x < 2 on a number line. Answer: An open circle at 2, with shading to the left.

9. How to help at home without turning it into memorisation

Begin with everyday language: “You must be at least 12 years old,” “The bag can weigh no more than 10 kg,” or “The temperature must stay below 5°C.” Ask your child to write the inequality before doing any calculations.

Next, draw a number line and test values on either side of the boundary. Only after this makes sense should you introduce algebraic rearrangement. Finally, practise one negative-division example and ask the child to explain why the direction changes.

A student who can justify the number line and check a boundary has a stronger foundation for GCSE inequalities than a student who merely remembers a sequence of symbol changes.

Next step: If your child is comfortable solving equations but struggles to represent all possible answers, ask for a short diagnostic assessment of inequality language, negative-number reasoning and number-line interpretation.

Official curriculum sources: Department for Education — National curriculum in England: mathematics programmes of study, Key Stages 3 and 4 and Department for Education — Mathematics guidance, Key Stage 3. The worked examples and practice questions are original, not reproduced exam questions.

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