AQA GCSE Mathematics Paper 2 (Higher), June 2025: Question 20

Identifying Errors in Number Line Inequalities

📋 What this question tests

This question tests your understanding of strict vs non-strict inequalities and how they are correctly represented on a number line. You must be able to recognise that strict inequalities (< or >) require open (unshaded) circles, and identify counter-examples where boundary values fail the inequality.

Question 20 • 1 Mark

David's Quadratic Inequality Solution

Analysis of David's representation of x² < 16

✅ Acceptable Correct Reasons (1 Mark)

Any one of the following clear reasons scores the mark:

Circle shading: "The circles should be open / unshaded / hollow / white."
Counter-example for 4: "4 is not a solution (because 4² = 16, and 16 is not less than 16)."
Counter-example for -4: "-4 is not a solution (because (-4)² = 16, which is not < 16)."
Diagram modification: Crossing out the solid dots and replacing them with empty/open circles at -4 and 4.

💡 Key Knowledge: Circles on Number Lines

< or > (Strict inequalities): Use an open circle (○). The value itself is not included.
≤ or ≥ (Non-strict inequalities): Use a solid / shaded circle (●). The value itself is included.
David's diagram shows -4 ≤ x ≤ 4 , but the question specifies x² < 16 , which simplifies to -4 < x < 4 .

📐 Step-by-Step Solving of x² < 16

Find the critical values: Solve the equation x² = 16 → x = 4 and x = -4 .
Determine the region: The curve y = x² - 16 is below the x-axis between the roots. Therefore, -4 < x < 4 .
Check the boundary symbol: Since the inequality is strictly < (not ≤), neither -4 nor 4 can be included.
Plot on the number line: Draw open circles at -4 and 4 and join them with a straight line.

🧠 Exam Technique: Writing "Give a reason" Answers

Be direct: State exactly what is visually wrong (e.g., "The circles should not be filled in").
Avoid vague statements: Simply writing what a symbol means without pointing out the mistake will score 0 marks.
Don't contradict yourself: Adding an incorrect statement alongside a correct one (choice) loses the mark (e.g. "Circles should be open and at -3 and 3" scores 0).

❌ Common Errors from the Mark Scheme (Scores 0)

"Coloured dot means equal to" — Too vague; describes meaning rather than stating David's specific mistake.
"The circles are the wrong colour" — Unclear mathematical terminology.
"His solution is -4 ≤ x ≤ 4" — Merely reading David's diagram without explaining why it is wrong.
"4 squared is 16" — An incomplete statement that doesn't state why this makes the representation incorrect.
"One circle should be white" — Incorrect; both circles must be open.

📊 Mark Scheme Summary

Mark Awarded: B1 (Independent)
Awarded for any correct mathematical explanation showing that the endpoints -4 and 4 are excluded from the solution set, or that the circle representation denotes inclusion incorrectly.