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. On this page you will learn what inequality symbols mean, how to solve single and double inequalities, how to represent solutions on a number line, and the one rule that catches out most students. If you can already solve linear equations, you are well on your way to mastering inequalities.

What Is an Inequality?

An inequality is a mathematical statement that compares two expressions using one of four symbols:

< means "less than"
> means "greater than"
≤ means "less than or equal to"
≥ means "greater than or equal to"

Unlike an equation, which has one solution (or a fixed number of solutions), an inequality typically has infinitely many solutions. For example, x > 3 is satisfied by 4, 5, 3.1, 100, and every other number greater than 3.

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.

Number Line Notation

An open circle (○) means the boundary value is not included (strict inequality: < or >).
A filled circle (●) means the boundary value is included (≤ or ≥).
Draw an arrow or a line in the direction of the valid values.

Step-by-Step Method

Solving a single linear inequality

Treat it like an equation. Perform the same inverse operations on both sides.
If you multiply or divide by a negative number, flip the inequality sign. This is the only extra rule.
Write the solution in the form x > a, x ≤ b, etc.
Represent on a number line if the question asks for it.

Solving a double inequality (e.g., 3 < 2x + 1 ≤ 11)

Perform operations on all three parts at the same time.
Aim to isolate x in the middle.
The result gives a range, for instance 1 < x ≤ 5.

Listing integer solutions

Some questions ask you to "list the integer values" that satisfy an inequality. After solving, simply write out every whole number in the valid range.

Worked Example 1 — Foundation Level

Question: Solve 5x − 2 > 13 and represent the solution on a number line.

Working:

Step 1: Add 2 to both sides. 5x > 15

Step 2: Divide both sides by 5. x > 3

Number line: draw an open circle at 3 and shade/arrow to the right.

Answer: x > 3

Worked Example 2 — Higher Level

Question: Solve −3(2x − 4) ≥ 30.

Working:

Step 1: Expand the bracket. −6x + 12 ≥ 30

Step 2: Subtract 12 from both sides. −6x ≥ 18

Step 3: Divide both sides by −6. Because we are dividing by a negative number, reverse the inequality sign. x ≤ −3

Check: Try x = −4 (which should satisfy x ≤ −3): −3(2(−4) − 4) = −3(−12) = 36. Is 36 ≥ 30? Yes ✓

Try x = 0 (which should not satisfy): −3(−4) = 12. Is 12 ≥ 30? No ✓

Answer: x ≤ −3

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.

Practice Questions

Q1 (Foundation): Solve 3x + 4 ≤ 19.

Answer: x ≤ 5

Q2 (Foundation/Higher): Solve the double inequality −1 < 2x + 3 ≤ 9 and list the integer values of x.

Answer: −2 < x ≤ 3, so the integer values are −1, 0, 1, 2, 3

Q3 (Higher): Solve 4 − 5x > 29.

Answer: x < −5

Need more help with inequalities? Start revising with GCSEMathsAI — our AI tutor walks you through each step and adjusts the difficulty as your confidence grows.

For a complete overview of what to revise, see our GCSE Maths Topics Complete List.

Summary

Inequalities compare two expressions using <, >, ≤, or ≥.
Solve them using the same method as linear equations.
The key extra rule: if you multiply or divide by a negative, reverse the inequality sign.
On a number line, use an open circle for strict inequalities and a filled circle for "or equal to."
Double inequalities require you to operate on all three parts simultaneously.
Always check your solution by substituting a value from inside your range and one from outside.
Listing integer solutions is a common exam question — count carefully and include or exclude boundaries as appropriate.

Solving Inequalities –