Every inequalities question from 15 Cambridge IGCSE Mathematics (0580) past papers (May/June 2025 to May/June 2026), syllabus section 2.6. 15 questions covering Core and Extended tiers, ordered from Core to Extended. Each question is labelled with its question paper code (for example 0580/12/M/J/26) and every final answer matches the official Cambridge mark scheme. Try each question first, then click “Show solution” to check your working. This page is part of our IGCSE Maths 0580 Algebra and Graphs past paper questions.
0580 Inequalities past paper questions: Core (Papers 1 and 3)
May/June 2026 · Paper 1 · 0580/12/M/J/26 · Q20
Write down the inequality represented on the number line: an open circle at −4, a solid circle at 5, joined by a line.
Show solution
- An open circle means the number is not included (<). A filled circle means it is included (≤).
- The open circle is at −4 and the filled circle is at 5, with x between them.
- −4 < x ≤ 5
May/June 2026 · Paper 3 · 0580/31/M/J/26 · Q18
(a) Write down the smallest integer that satisfies x > −8.
(b) Represent the inequality −4 ≤ x < 3 on the number line.
Show solution
- x > −8 means x is bigger than −8, and −8 itself is not allowed.
- The next integer bigger than −8 is −7
- −4 ≤ x means −4 is included: filled circle. x < 3 means 3 is not included: open circle.
- Draw a filled circle at −4 and an open circle at 3, joined by a line.
May/June 2026 · Paper 3 · 0580/33/M/J/26 · Q14
Represent the inequality n < 5 on the number line.
Show solution
- n < 5 means 5 itself is not allowed, so draw an open (empty) circle at 5.
- n can be any number smaller than 5, so draw an arrow pointing left.
- open circle at 5 with an arrow pointing left
Oct/Nov 2025 · Paper 1 · 0580/13/O/N/25 · Q13
Write down the inequality represented on the number line: a filled circle at −5 and an open circle at 1, joined by a line.
Show solution
- A filled circle means the number is included (≤). An open circle means it is not included (<).
- The filled circle is at −5 and the open circle is at 1.
- −5 ≤ x < 1
Oct/Nov 2025 · Paper 3 · 0580/31/O/N/25 · Q8
Solve. (a) 8x = 32 (b) 6x − 3 = 12 (c) Represent the inequality −4 ≤ x < 2 on the number line.
Show solution
- Divide both sides by 8: x = 32 ÷ 8.
- x = 4
- Add 3 to both sides: 6x = 15.
- Divide by 6: x = 2.5
- −4 ≤ x means −4 is included: filled circle. x < 2 means 2 is not included: open circle.
- Draw a filled circle at −4 and an open circle at 2, joined by a line.
Oct/Nov 2025 · Paper 3 · 0580/32/O/N/25 · Q17
Represent the inequality x ≥ 3 on the number line.
Show solution
- x ≥ 3 means 3 is allowed, so draw a filled (solid) circle at 3.
- x can be any number bigger than 3, so draw an arrow pointing right.
- filled circle at 3 with an arrow pointing right
May/June 2025 · Paper 1 · 0580/12/M/J/25 · Q19
(a) P is a prime number. Write down the value of P that satisfies the inequality 13 < P < 19.
(b) Write down the inequality represented on the number line.
Show solution
- The whole numbers between 13 and 19 are 14, 15, 16, 17 and 18.
- 14, 16 and 18 are even, and 15 = 3 × 5. Only 17 is prime.
- P = 17
- A filled circle means the number is included (≤ or ≥). An open circle means it is not included (< or >).
- The filled circle is at −2 and the open circle is at 7, with the line between them.
- −2 ≤ x < 7
May/June 2025 · Paper 1 · 0580/13/M/J/25 · Q15
(a) Write down the inequality represented on the number line.
(b) Write down the smallest integer that satisfies the inequality d > −31⁄4.
Show solution
- A filled circle means “included” (≤). An open circle means “not included” (<).
- The filled circle is at −4 and the open circle is at 3.
- −4 ≤ x < 3
- −31⁄4 = −3.25, and d must be greater than −3.25.
- On a number line, the numbers to the right of −3.25 are bigger. The first integer to the right is −3.
- Smallest integer = −3
- Tip: −4 is not allowed, because −4 is less than −3.25.
0580 Inequalities past paper questions: Extended (Papers 2 and 4)
May/June 2026 · Paper 2 · 0580/22/M/J/26 · Q12
(a) On the grid, draw the straight lines x = 8, y = 2, x + y = 8 and 2y − x = 4.
(b) Region R satisfies x ≤ 8, y ≥ 2, x + y ≥ 8 and 2y − x ≤ 4. By shading the unwanted regions of the grid, find and label the region R.
Show solution
- x = 8 is a vertical line through 8 on the x-axis. y = 2 is a horizontal line through 2 on the y-axis.
- For x + y = 8, find two points: x = 0 gives y = 8, and y = 0 gives x = 8. Join (0, 8) and (8, 0).
- For 2y − x = 4: x = 0 gives y = 2, and x = 8 gives 2y = 12, so y = 6. Join (0, 2) and (8, 6).
- All four inequalities use ≤ or ≥, so draw all four as solid lines.
- Test a point to decide which side to keep for each line, then shade the side you do not want.
- Keep: left of x = 8, above y = 2, above x + y = 8 and below 2y − x = 4.
- The unshaded region left over is R.
- R is the quadrilateral with corners (4, 4), (6, 2), (8, 2) and (8, 6)
May/June 2026 · Paper 2 · 0580/23/M/J/26 · Q3
Write down the inequality shown by the number line.
Show solution
- A filled circle means the number is included (≤). An open circle means it is not included (<).
- The filled circle is at −1 and the open circle is at 5.
- −1 ≤ x < 5
Feb/Mar 2026 · Paper 2 · 0580/22/F/M/26 · Q8
Find the integer values of x that satisfy −3 ≤ x < 1.
Show solution
- −3 ≤ x means −3 is included. x < 1 means 1 is not included.
- List the integers from −3 up to (but not including) 1.
- −3, −2, −1, 0
Feb/Mar 2026 · Paper 4 · 0580/42/F/M/26 · Q13
The region marked R on the grid is defined by three inequalities. Find the three inequalities.
Show solution
- For each boundary line, find its equation, then decide which side R is on.
- Broken line means < or > (not included). Solid line means ≤ or ≥ (included).
- Vertical solid line x = 3, with R to the right: x ≥ 3
- Line through (1, 0) and (3, 2): gradient = 2 ÷ 2 = 1, and it crosses the y-axis at −1, so y = x − 1. It is broken and R is below it: y < x − 1
- Line through (0, 2) and (4, 0): gradient = −2 ÷ 4 = −1⁄2, intercept 2, so y = −1⁄2x + 2. It is solid and R is above it: y ≥ −1⁄2x + 2
- Tip: To check a side, pick a point inside R and test it in each inequality.
Oct/Nov 2025 · Paper 2 · 0580/21/O/N/25 · Q16
Write down all the inequalities that define the region R shown on the grid.
Show solution
- Find the equation of each boundary line, then decide which side of it R is on.
- A solid line means the boundary is included (≤ or ≥). A broken line means it is not included (< or >).
- The vertical line at x = 2.5 is solid and R is to its left: x ≤ 2.5
- The horizontal line y = 3 is broken and R is above it: y > 3
- The horizontal line y = 4 is solid and R is below it: y ≤ 4
- The sloping line through (0, 0) and (2, 4) is y = 2x. It is solid and R is below it: y ≤ 2x
Oct/Nov 2025 · Paper 2 · 0580/22/O/N/25 · Q7
Represent the inequality −4 < x ≤ 3 on the number line.
Show solution
- −4 < x means −4 is not included, so draw an open circle at −4.
- x ≤ 3 means 3 is included, so draw a filled circle at 3.
- Join the circles with a line: open circle at −4 and a filled circle at 3
Oct/Nov 2025 · Paper 4 · 0580/41/O/N/25 · Q7
(a) Write down the inequality represented on the number line: an open circle at −3 and a filled circle at 2, joined by a line.
(b) Write down the integer values of x that satisfy the inequality −4 < 2x ≤ 8.
Show solution
- An open circle means the number is not included, so x is bigger than −3: −3 < x.
- A filled circle means the number is included, so x is less than or equal to 2: x ≤ 2.
- Put them together: −3 < x ≤ 2
- Start with −4 < 2x ≤ 8.
- Divide every part by 2: −2 < x ≤ 4.
- x is bigger than −2 (so −2 is not allowed) and can be 4.
- The integers are −1, 0, 1, 2, 3, 4
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