Sheet № 15 · Foundation + Higher · AQA · Edexcel · OCR
Solving Inequalities –
Inequalities work almost exactly like equations — with one crucial twist. Instead of finding a single value that satisfies an equals sign, you find a range of values that satisfy an inequality sign. This topic appears on both Foundation and Higher tier papers for AQA, Edexcel, and OCR, and it is closely linked to solving linear equations.
§Key definitions
Question:
Solve 5x − 2 > 13 and represent the solution on a number line.
Check:
Try x = −4 (which should satisfy x ≤ −3): −3(2(−4) − 4) = −3(−12) = 36. Is 36 ≥ 30? Yes ✓
Q1 (Foundation):
Solve 3x + 4 ≤ 19.
Q2 (Foundation/Higher):
Solve the double inequality −1 < 2x + 3 ≤ 9 and list the integer values of x.
Q3 (Higher):
Solve 4 − 5x > 29.
§Formulas to memorise
If you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign.
Solving inequalities uses the same inverse operations as solving equations, with the sign-flip rule above as the only difference.
< — means "less than"
> — means "greater than"
≤ — means "less than or equal to"
≥ — means "greater than or equal to"
An open circle (○) means the boundary value is not included (strict inequality: < or >).
Treat it like an equation. — Perform the same inverse operations on both sides.
Write the solution — in the form x > a, x ≤ b, etc.
Represent on a number line — if the question asks for it.
Worked example
Solve 5x − 2 > 13 and represent the solution on a number line.
Working:
⚠ Common mistakes
- ✗Forgetting to flip the inequality sign when multiplying or dividing by a negative number. This is by far the most common error. Practise it until it becomes automatic.
- ✗Using the wrong circle on the number line. Open circle for < and >, filled circle for ≤ and ≥. If you mix these up you will lose the mark.
- ✗Listing integers outside the range. For −2 < x ≤ 3 where x is an integer, the values are −1, 0, 1, 2, 3. Note that −2 is not included (strict inequality) but 3 is included.
- ✗Writing the inequality the wrong way round. If you get x > 3 but write 3 > x, you have reversed the meaning. Always keep x on the left or re-read your solution to check direction.
- ✗Treating inequalities as equations and writing "=". Keep the inequality symbol throughout your working.
✦ Exam tips
- →Double inequalities are worth practising. They appear frequently on both tiers and are often worth 3 marks. Subtract, then divide, keeping all three parts aligned.
- →On "list the integers" questions, count carefully. Examiners award the mark only if every correct integer is listed and no extras are included.
- →If the question says "show on a number line," you must draw one — a written inequality alone will not earn the marks.
- →AQA and Edexcel sometimes combine inequalities with graphs at Higher level. You may need to shade a region satisfying multiple inequalities. Practise identifying which side of a line to shade.