Inequalities
Solving and representing inequalities on a number line and in regions.
Review these first
Learning objectives
What you need to be able to do
Teacher-mapped phrasing — check against the official Cambridge syllabus for exact wording.
- 2.4.1Solve linear inequalities and represent solutions on a number line.
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Inequalities
An inequality is solved almost exactly like an equation, with one critical exception.
The symbols
<less than,>greater than≤less than or equal to,≥greater than or equal to
The one rule that differs
Multiplying or dividing by a negative number reverses the inequality sign. −2x > 6 → divide by −2 → x < −3 Sanity check with a number: x = −4 gives −2(−4) = 8, which is indeed greater than 6 ✓ Many students avoid the problem entirely by rearranging so the variable ends up positive: −2x > 6 becomes −6 > 2x, so x < −3 with no sign flip required.
Number line notation
- Open circle ○ for
<or>— the endpoint is not included. - Filled circle ● for
≤or≥— the endpoint is included.
Double inequalities
−3 < x ≤ 5 means x is between −3 and 5, excluding −3 but including 5. Operate on all three parts at once.
Integer solutions
If asked to list integers satisfying −2 < x ≤ 3, they are −1, 0, 1, 2, 3. Check each endpoint carefully against whether the inequality is strict.
Think of it like this
Reversing the sign when dividing by a negative is like turning around while walking: everything that was ahead is now behind. The relationship is unchanged, but the direction you describe it from has flipped.
Worked examples
Method, step by step
Solve 5 − 2x ≥ 11 and show the solution on a number line.
- 1Subtract 5 from both sides: −2x ≥ 6.
- 2Divide by −2 and reverse the sign: x ≤ −3.
- 3Because it is ≤, use a filled circle at −3 with an arrow pointing left.
- 4Check: x = −4 gives 5 − 2(−4) = 13 ≥ 11 ✓
x ≤ −3, shown with a filled circle at −3 and an arrow to the left
Common misconceptions
- Forgetting to reverse the sign when multiplying or dividing by a negative — the single biggest source of lost marks here.
- Using a filled circle for a strict inequality. `<` and `>` require an open circle.
- Including an excluded endpoint when listing integers, e.g. counting −2 in `−2 < x`.
In the exam
- Test your final answer with one value from the solution set. It confirms the direction of the inequality immediately.
- When listing integers, write the range first, then read off the values — it avoids endpoint slips.