Mathematics 0580 · syllabus C2.6, E2.6
Cambridge IGCSE Mathematics 0580 — Inequalities topical past papers
Showing inequalities on a number line, and on the Extended route solving linear inequalities and shading or describing regions on a graph.
- Questions
- 112
- Years
- 2009–2026
- With mark scheme
- 88%
- Questions 2022–2026
- 38
Anyone can browse these pages. Opening the questions and their mark schemes takes a free Momentum account.
What Inequalities covers
An inequality says that one quantity is bigger or smaller than another, using <, >, ≤ and ≥. Core candidates show inequalities on a number line and read them from one: an open circle marks a strict inequality (< or >) and a filled circle an inclusive one (≤ or ≥), so −2 < x ≤ 3 runs from an open circle at −2 to a filled circle at 3. Questions also ask for the integers that satisfy an inequality, which for this one are −1, 0, 1, 2 and 3.
Extended candidates construct and solve linear inequalities, such as 4x − 7 > 2x + 3 or −5 ≤ 2x + 1 < 9, with the same steps as equations, except that multiplying or dividing by a negative number reverses the sign. They also show inequalities in x and y as regions on a graph: draw each boundary line, broken for < or > and solid for ≤ or ≥, and shade the unwanted side unless the question says otherwise. The reverse task is to write down the inequalities that describe a region already drawn.
Number-line inequalities are Core content, so they can be set on all four papers; solving inequalities and drawing regions are Extended, examined only on Papers 2 and 4. Linear programming has left the syllabus, but older linear programming questions still practise drawing and describing regions; skip any part that asks for the greatest or least value of a quantity in the region. Inequalities draw on solving equations, and regions on the straight-line graphs of coordinate geometry.
In the syllabus
The headings this topic has in the Cambridge International 2025–2027 syllabus, as the board prints them.
- C2.6 Inequalities
- E2.6 Inequalities
What you should be able to do
- Show an inequality on a number line with open and filled circles, and write down the inequality a number line shows
- List the integers that satisfy an inequality or a pair of inequalities
- Solve linear inequalities, including double inequalities (Extended)
- Draw the region defined by inequalities in x and y, with broken or solid boundary lines (Extended)
- Write down the inequalities that describe a region drawn on a graph (Extended)
Key ideas
- x < a and x > a are strict; x ≤ a and x ≥ a include a itself
- Adding or subtracting, or multiplying or dividing by a positive number, keeps the sign; multiplying or dividing by a negative number reverses it
- A boundary line is broken for < or > and solid for ≤ or ≥
- Testing a point off the line, such as (0, 0), shows which side of a boundary satisfies the inequality
How Inequalities has been examined
From 2022 to 2026, Inequalities was on Paper 4 · Calculator (Extended) in 9 of 14 exam series, worth about 2% of that paper’s marks.
| Paper | Questions | Share of marks | Series |
|---|---|---|---|
| Paper 1 · Non-calculator (Core) | 7 | 1% | 6 of 14 |
| Paper 2 · Non-calculator (Extended) | 12 | 2% | 9 of 14 |
| Paper 3 · Calculator (Core) | 7 | 1% | 5 of 14 |
| Paper 4 · Calculator (Extended) | 12 | 2% | 9 of 14 |
Momentum’s own count, over the questions in its bank for these years: a series is one exam session, every zone included, and a question filed under two chapters counts in both. Median marks per question: 3.
Filed in Momentum under: Solving Inequalities, Linear Programming.
A worked example
Solve −5 ≤ 3x + 4 < 13, and list the integers that satisfy the inequality.
- Subtract 4 from all three parts: −9 ≤ 3x < 9.
- Divide all three parts by 3, which is positive, so the signs stay the same: −3 ≤ x < 3.
- x = −3 is included (≤) and x = 3 is not (<).
- So the integers are −3, −2, −1, 0, 1 and 2.
- Check the ends: x = −3 gives 3(−3) + 4 = −5, which is allowed, and x = 3 gives 13, which is not less than 13.
Answer: −3 ≤ x < 3; the integers are −3, −2, −1, 0, 1 and 2
Written by Momentum for this page, with its own numbers; it is not taken from a past paper.
Mistakes that cost marks
- Using a filled circle for a strict inequality: x > 2 does not include 2, so the circle at 2 must be open.
- Dividing by a negative number without reversing the sign: −3x < 12 gives x > −4, not x < −4.
- Shading the region you want: the convention is to shade the unwanted side of each line unless the question says otherwise, so the answer region is the one left clear.
- Drawing a solid line for a strict inequality, or a broken one for an inclusive inequality: the style of the line is part of the answer.
- Getting the end points wrong when listing integers: for −2 < x ≤ 3, −2 is not included but 3 is.
Exam technique
- Solve an inequality exactly as you would an equation, keeping the inequality sign throughout and reversing it whenever you multiply or divide by a negative number.
- For regions, draw each boundary from two points, label it, and test (0, 0) or another clear point to decide which side to shade.
- When listing integers, write out the end points and decide for each one whether it is included.
Practising in Momentum
- Every question in a chapter as one list, from every paper and year in the bank, newest first or shuffled.
- Each question turns over to its own part of the mark scheme, where the board published one.
- Narrow by paper, year, session, zone and marks, and star the questions worth keeping.
- Pick chapters, a number of questions and a time, and the exam builder draws a mock to sit and mark.
Past-paper questions and mark schemes are the copyright of Cambridge International. Momentum shows them to signed-in learners for revision and does not publish them on these pages.