AQA A-Level Mathematics Paper 2, June 2025: Question 3

1 mark · Easy difficulty · Short Answer

Identify the set notation representing the solution to the quadratic inequality $(x + 2)(x - 7) > 0$ using the provided graph.

Practise this question

Question

A graph of the quadratic equation y = (x + 2)(x - 7) showing a U-shaped parabola intersecting the x-axis at -2 and 7, with the vertex below the x-axis. The question asks to solve the inequality (x + 2)(x - 7) > 0 by ticking one box from four options written in set notation involving open and closed intervals.
Question text

3 The diagram shows the graph with equation y = (x + 2)(x – 7)

y

–2 O 7 x

Solve the inequality.

(x + 2)(x – 7) > 0

Tick ( ) one box.

[1 mark]

x ∈ (–∞, –2) ∩ (7, ∞)

∩

x ∈ (–∞, –2) (7, ∞)

x ∈ (–∞, –2] ∩ [7, ∞)

∩

x ∈ (–∞, –2] [7, ∞)

Mark scheme

Show the mark scheme Mark scheme for Question 3 showing 1 mark (B1) for ticking the second box, with typical solution x in (-infinity, -2) union (7, infinity).

Q Marking instructions AO Marks Typical solution

3 Ticks second 2.5 B1 x − −(,2) (7, )

Question 3 Total 1

How to answer it

Solving Quadratic Inequalities in Set Notation

What this question tests
  • Interpreting Quadratic Graphs: Reading regions above the x-axis where y > 0.
  • Set & Interval Notation: Correct use of open parentheses ( ) versus closed brackets [ ] for strict inequalities.
  • Set Operations: Understanding the difference between Union ( ∪ ) and Intersection ( ∩ ) when defining disjoint solution regions.
Question 3 (1 Mark)

Identify the Correct Solution Set for (x + 2)(x − 7) > 0

AQA A-Level Mathematics • Assessment Objective 2.5

📐 Step-by-Step Analysis

  1. Identify Critical Values:
    The graph crosses the x-axis at x = -2 and x = 7 .
  2. Identify the Region:
    The inequality requires (x + 2)(x − 7) > 0 , which means y > 0 (strictly above the x-axis). From the curve, this occurs when x < -2 or x > 7 .
  3. Choose Interval Brackets:
    Because it is a strict inequality ( > and not ≥ ), endpoints -2 and 7 are excluded. Round brackets ( ) must be used: (-∞, -2) and (7, ∞) .
  4. Select Set Operator:
    A number cannot be simultaneously less than -2 and greater than 7 . The solution consists of values in the first interval OR the second interval, meaning we need the Union symbol ( ∪ ).

✅ Correct Answer

Box 2 (the second box)

x ∈ (-∞, -2) ∪ (7, ∞)

Mark Scheme:
B1 for correctly selecting the second box. (AO 2.5)

💡 Key Knowledge

  • Parentheses vs Brackets:
    • (a, b) excludes endpoints (used with < , > , and ±∞ ).
    • [a, b] includes endpoints (used with ≤ and ≥ ).
  • Set Operators:
    • ∪ (Union) means "OR" — either set satisfies the condition.
    • ∩ (Intersection) means "AND" — only values belonging to both sets simultaneously.

🧠 Exam Technique & Strategy

  • Eliminate Impossible Sets: Disjoint intervals (like numbers below -2 and above 7) have no overlap. Therefore, (-∞, -2) ∩ (7, ∞) = ∅ (the empty set). Any option containing an intersection symbol ( ∩ ) can be ruled out immediately!
  • Check the Inequality Sign: Look closely at whether the question has > or ≥ to immediately decide between round ( ) and square [ ] brackets.

❌ Common Traps & Misconceptions

  • Confusing ∩ and ∪: Choosing Option 1 by mistake. Remember that intersection ( ∩ ) requires values to satisfy both at once, which is impossible here.
  • Using Square Brackets: Choosing Option 4 ( [-∞, -2] ∪ [7, ∞) ). Square brackets include the boundary points, which would solve (x + 2)(x - 7) ≥ 0 , not > 0 . Furthermore, infinity ( ∞ ) can never take a square bracket.
  • Inside vs Outside Regions: Thinking the solution is the single interval between the roots, which would be (-2, 7) , representing y < 0 .

Topics

Pure Mathematics · B: Algebra and functions

Question and mark scheme from the AQA A-Level Mathematics examination, Paper 2, June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.