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Inequalities on graphs

Free Printable Inequalities on graphs PDF Worksheets, Revision Guide & GCSE Past Paper Questions (Grade 6)

Download free inequalities on graphs worksheets (PDF), GCSE maths past paper questions and worked solutions for Inequalities on graphs (6G). This is a Grade 6 algebra topic from the GCSE questions by topic collection for Higher tier GCSE Maths (Edexcel, AQA, OCR and IGCSE).

The free PDF worksheets table below brings together printable question booklets (Qs) and worked answers or mark schemes (Ans) from Maths Genie, Corbett Maths, Maths Made Easy, Dr Frost, Naiker Maths, 1st Class Maths, Metatutor, Maths Takeaway, Dr Austin, Maths4EveryOne, E Classroom and PMT, along with past paper question booklets.

Below the worksheet links you can read our inequalities on graphs revision notes and worked examples with diagrams, try real GCSE past paper inequalities on graphs questions with step-by-step solutions, or download a PDF of all inequalities on graphs worksheet links.

Free Inequalities on graphs PDF Worksheets & Answers

Grade 6 • Topic 6G
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GCSE 9–1 practice booklet & solutions
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PMT (Edxl)
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Grade 6 topics
6A - Recurring decimals6B - Harder indices
6C - The product rule6D - Repeated change
6E - Triple brackets6F - Parallel and perpendicular
6G - Inequalities on graphs6H - Similar shapes (area and vol)
6J - Negative scale factors6K - Circle theorems
6L - Cumulative frequency6M - Box plots
6N - Combined transformations6P - Capture recapture
GCSE Maths Grade 6 Revision Notes & Method

Guide to Inequalities on graphs: Explanation, Diagrams & Worked Examples

Download Printable One Page Guide (PDF)

Explanation:

To represent inequalities on a graph, first draw each boundary line by replacing the inequality sign with =. Use a solid line for inclusive inequalities (≤ or ≥) and a dashed line for strict inequalities (< or >).

To decide which side of a line satisfies the inequality, substitute a simple test point not on the line—such as (0, 0) or (1, 1)—into the inequality and check if it is true.

Example 1: Write down the three inequalities that define the region R (containing (2, 0)) bounded by solid lines x = 1 and y = −2, and dashed line x + y = 4.

R x = 1 y = −2 x + y = 4

Solution:

Right of solid x = 1 → x ≥ 1; above solid y = −2 → y ≥ −2
Below dashed x + y = 4 (test (2,0): 2 + 0 < 4) → x + y < 4
Answer: x ≥ 1, y ≥ −2, and x + y < 4

Example 2: Find all integer coordinates (x, y) that satisfy x > 1, y > 2, and 2x + y ≤ 7.

Solution:

Since x and y are integers: x ≥ 2 and y ≥ 3
Try x = 2: 2(2) + y ≤ 7 → y ≤ 3, so y = 3 works → (2, 3)
Try x = 3: 2(3) + y ≤ 7 → y ≤ 1 (no integers with y > 2)
Answer: (2, 3)
Scanned GCSE Exam Questions & Worked Solutions

GCSE Past Paper Inequalities on graphs Questions & Answers

Past Paper Example 1 GCSE Jun 2024 2H Q13 3 marks
Full Paper (PDF) • Mark Scheme (PDF)
xy–6–5–4–3–2–1123456–6–5–4–3–2–1123456O
On the grid, shade the region R that satisfies all the following inequalities.
x ≤ 2   y ≥ -3   y ≤ 2x + 1   3x + 2y ≤ 6
Label the region R.
On the grid, shade the region R that satisfies all the following inequalities.
x ≤ 2   y ≥ -3   y ≤ 2x + 1   3x + 2y ≤ 6
Label the region R.
Worked Solution & Mark Scheme (3 marks)
Plot boundary lines x = 2, y = –3, y = 2x + 1 (through (–2,–3), (0,1), (2,5)), and 3x + 2y = 6 (through (0,3), (2,0)) M2
Shade interior region R. A1
xyR–6–5–4–3–2–1123456–6–5–4–3–2–1123456O
Answer: Quadrilateral region R bounded by x = 2, y = –3, y = 2x + 1 and 3x + 2y = 6 with vertices (–2, –3), (2, –3), (2, 0) and (6/7, 19/7)
Past Paper Example 2 GCSE Nov 2017 2H Q14 3 marks
Full Paper (PDF) • Mark Scheme (PDF)
On the grid, shade the region that satisfies all these inequalities.
y > 1 x + y < 5 y > 2x
Label the region **R**.
–6–5–4–3–2–11234567–6–5–4–3–2–1123456Oxy
Worked Solution & Mark Scheme (3 marks)
Draw lines y = 1, x + y = 5 and y = 2x M2
shade the region above y = 1, below x + y = 5 and to the left of y = 2x. A1
R–6–5–4–3–2–11234567–6–5–4–3–2–1123456Oxy
Answer: Region R shaded above y = 1, below x + y = 5 and above/left of y = 2x
Past Paper Example 3 GCSE Nov 2021 1H Q17 3 marks
Full Paper (PDF) • Mark Scheme (PDF)
xy–6–5–4–3–2–1123456–6–5–4–3–2–1123456O
On the grid show, by shading, the region that satisfies all of these inequalities.
2y + 4 < x   x < 3   y < 6 - 3x
Label the region **R**.
Worked Solution & Mark Scheme (3 marks)
Draw boundary lines x = 3, y = 0.5x – 2, and y = 6 – 3x M2
Shade the region satisfying all three inequalities. A1
xyR–6–5–4–3–2–1123456–6–5–4–3–2–1123456O
Answer: Region R shaded below 2y + 4 = x, below y = 6 – 3x, and left of x = 3

← 6F Parallel and perpendicular

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6H Similar shapes (area and vol) →