AQA GCSE Mathematics Paper 3 (Higher), June 2025: Question 19

4 marks · Medium difficulty · Multi-step Problem

Identify and label region R on the grid defined by the inequalities x + y < 5, y < 2x + 4, and y ≥ 1.

Practise this question

Question

A Cartesian coordinate grid with both x-axis and y-axis extending from -6 to 6 with increments of 1. Above the grid, the question asks to identify the region represented by x + y < 5 and y < 2x + 4 and y ≥ 1, and to label the region R.

Mark scheme

Show the mark scheme Mark scheme for question 19. It gives B4 for drawing x + y = 5 and y = 2x + 4 as dashed lines, y = 1 as a solid line, and correctly identifying region R. Partial credit is given: B1 for one line drawn, B2 for two lines drawn, and B3 for all three lines drawn as either dashed or solid. A diagram illustrates the three boundary lines intersecting to enclose a triangular region labelled R.

How to answer it

Representing Inequalities on a Coordinate Grid

📌 What this question tests

Plotting linear boundary equations ( y = mx + c , intercept form x + y = k , and horizontal lines y = c ), distinguishing correctly between strict inequalities (dashed lines) and non-strict inequalities (solid lines), and accurately identifying the bounded enclosed region that satisfies all three inequalities simultaneously.

Question 19 (4 Marks)

Linear Programming: Identifying Region R

Inequalities given: x + y < 5 ,   y < 2x + 4 ,  and  y ≥ 1

📐 Step-by-Step Boundary Lines

  1. Line 1: x + y = 5
    • Find intercepts: when x = 0, y = 5; when y = 0, x = 5.
    • Inequality is < (strict): draw a dashed line through (0, 5) and (5, 0).
  2. Line 2: y = 2x + 4
    • Intercept at (0, 4), gradient = 2. Plots through (-2, 0), (0, 4), and (1, 6).
    • Inequality is < (strict): draw a dashed line.
  3. Line 3: y = 1
    • Horizontal line passing through all points where y = 1.
    • Inequality is ≥ (inclusive): draw a solid line.
  4. Region Test: Try (0, 2)
    • 0 + 2 = 2 < 5 (True)
    • 2 < 2(0) + 4 = 4 (True)
    • 2 ≥ 1 (True)
    The triangular region containing (0, 2) is Region R.

💡 Key Knowledge

  • Line Styles:
    • Strict ( < or > ) = Dashed line (points on the line are excluded).
    • Inclusive ( ≤ or ≥ ) = Solid line (points on the line are included).
  • Which side to choose?
    • y < 2x + 4 means the region below the line.
    • x + y < 5 (or y < 5 - x ) means the region below/left of the line.
    • y ≥ 1 means the region above the line.

✅ Final Identification of Region R

The bounded region forms a triangle enclosed by the three vertices:

  • Intersection of y = 1 and y = 2x + 4 : at (-1.5, 1)
  • Intersection of y = 1 and x + y = 5 : at (4, 1)
  • Intersection of y = 2x + 4 and x + y = 5 : at (0.33, 4.67)

Place a clear, unambiguous label R inside this triangular space.

🧠 Exam Technique & Mark Scheme Breakdown

  • B1: 1 boundary line correctly drawn (solid or dashed).
  • B2: 2 boundary lines correctly drawn (solid or dashed).
  • B3: All 3 boundary lines correctly drawn (solid or dashed).
  • B4: All 3 lines drawn with correct line styles (2 dashed, 1 solid) AND region R correctly identified.
  • Tip: Always test a point inside your selected area (e.g. (1, 2) ) against all 3 inequalities to confirm before labelling.

❌ Common Errors & Pitfalls

  • Drawing all lines as solid: You immediately lose the final mark (stuck on 3/4) if you do not use dashed lines for < .
  • Confusing y = 1 with x = 1 : Remember that y = 1 is a horizontal line cutting the vertical axis at 1, while x = 1 is vertical.
  • Incorrect line plotting: Drawing y = 2x + 4 with a gradient of 1/2 instead of 2 (i.e. going across 2, up 1 rather than across 1, up 2).
  • Shading ambiguity: If you shade out rejected areas, make sure the remaining unshaded region is clearly labelled R so the examiner knows which region you intend.
Examiner Insight: Many candidates lose a mark simply because they draw every line solid using a ruler without checking the inequality symbols. Use a sharp pencil and ruler; draw dashed lines explicitly using clear breaks.

Topics

Algebra · 3.2.2 Graphs · 3.2.3 Solving equations and inequalities

Question and mark scheme from the AQA GCSE Mathematics examination, Paper 3 (Higher), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.