Pearson Edexcel International GCSE in Mathematics (Specification A) · 4MA1 · Free · no account needed

Inequalities

In one sentence. Solving inequalities like equations — with the one extra rule about negatives — and reading off the integers they allow.

Algebra — one of 8 modules in this strand, and one of 53 in Pearson Edexcel International GCSE Mathematics. Both tiers.

This module ends with Brahmagupta. Learn Inequalities, teach it, then defend it against Brahmagupta.

Where this sits in the specification

Specification 4MA1, 2.8 Inequalities. Incandio is aligned to this specification; it is not published by, endorsed by or affiliated with Pearson, and the sub-topic numbers are given so you can check every module against your own copy.

  • 2.8 — Inequalities

1 · Understand it

What it is

An inequality states a range rather than a value: 3x + 5 < 17 is solved exactly like an equation, giving x < 4 — every number below 4 works. Double inequalities (−3 < 2x + 1 ≤ 9) are two inequalities worn as one; whatever you do, do to all THREE parts.

Why it works

Balanced moves preserve order — with one exception. Multiplying or dividing by a NEGATIVE reverses every comparison (3 < 5, but −3 > −5: the number line flips through zero). That's why solving 2 − 3x > 11 either flips the symbol at the ÷(−3) step, or avoids the issue by moving the x-term to the positive side.

When a question is asking for it

'Solve the inequality' (keep the symbol, not an equals sign), 'find the integers satisfying…' (solve, then LIST), and 'the largest/smallest integer such that…' (solve, then step to the nearest allowed whole number).

How to hold it in your head

An inequality's answer is a beam of the number line: x < 4 is everything left of an open dot at 4; −2 < x ≤ 4 is a segment, open one end, filled at the other. Draw the beam — questions about integers become counting dots on a picture.

2 · The mistakes that cost marks

  • Replacing the inequality with '=' and answering x = 4. The symbol IS the answer's shape: x < 4.
  • Forgetting the flip: from −3x > 9, dividing by −3 gives x < −3 — the symbol turns around.
  • Treating strict (<) and inclusive (≤) as interchangeable when listing integers: x < 4 excludes 4; x ≤ 4 includes it.
  • Operating on only two parts of a double inequality. Subtract 1 from ALL THREE parts of −3 < 2x + 1 ≤ 9.

3 · Worked examples

Laid out the way an examiner wants working shown — every step is a mark.

Solve −3 < 2x + 1 ≤ 9.

  1. Subtract 1 everywhere: −4 < 2x ≤ 8.
  2. Divide everywhere by 2: −2 < x ≤ 4.

Answer: −2 < x ≤ 4

Solve 2 − 3x > 11.

  1. Subtract 2: −3x > 9.
  2. Divide by −3 AND FLIP the symbol: x < −3.
  3. (Check with x = −4: 2 − 3(−4) = 14 > 11 ✓)

Answer: x < −3

4 · Prove it — the five questions

The five questions climb Grade 6 → 9+ and are marked inside Incandio on your own device, against authored accepted forms, with a diagnosis of the wrong answer you actually gave. The answers stay in the app so that the practice is worth doing; the questions themselves are here.

  1. Grade 6 — Solve the inequality 3x + 5 < 17. (Type your answer like x<4.)
  2. Grade 7 — Work out the largest integer that satisfies 5x − 2 ≤ 24.
  3. Grade 8 — Solve −3 < 2x + 1 ≤ 9. (Type your answer like -2<x<=4.)
  4. Grade 9 — Solve 2 − 3x > 11. (Type your answer like x<-3.)
  5. 9+ — How many integers n satisfy n² ≤ 16?

Think you understand it?

Reading is the start. On Incandio a topic counts as mastered only once you have taught it and defended it — the two things a revision guide cannot ask of you.

Incandio this topic → Prove it. Teach it. Defend it. What does “Incandio it” mean?

  1. Prove it — five questions on Inequalities, marked on the spot, that tell you what was missing rather than just what was wrong.
  2. Teach it — Explain how Inequalities works — and why — to Ember, an AI pupil who does not have it yet. The method matters more than the vocabulary.
  3. Defend it — Argue Inequalities against Brahmagupta — three rounds, marked against descriptors you can read before you begin.
  4. Then the exam — read all 11 authored exam-style questions on this unit, each with its command word and its tariff, then have them marked on Incandio with the mistake you actually made named.

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