A6AlgebraFoundation & Higher

Inequalities

An inequality says one thing is bigger or smaller than another, rather than exactly equal. You solve them just like equations — with one important extra rule about negative numbers.

35 min Video by Maths Genie AQA GCSE Maths

Solving InequalitiesWatch the walkthrough, then read the notes below.

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What you'll learn

Read and write inequalities
Show inequalities on a number line
Solve linear inequalities
Know the rule for multiplying or dividing by a negative

The symbols and number lines

$<$ less than, $>$ greater than, $\leq$ less than or equal to, $\geq$ greater than or equal to. On a number line, an open circle means “not included” ( $<, >$ ) and a filled circle means “included” ( $\leq, \geq$ ).

Solving inequalities

Solve exactly like an equation using inverse operations. The one special rule: if you multiply or divide both sides by a $negative$ number, you must $flip$ the inequality sign.

1Subtract 1:

2 x < 8

2Divide by 2:

x < 4

Answer

x < 4

Tip — Only flip the sign when multiplying or dividing by a negative — not for adding or subtracting.

Remember these

< \leq > \geq

Less than, ≤, greater than, ≥.

\div (-) \Rightarrow flip the sign

Dividing by a negative flips the inequality.

Watch out for these

Forgetting to flip the sign when dividing by a negative.

- 2 x < 6

becomes

x > - 3

(sign flips).

Using a filled circle for a strict inequality.

Open circle for < and >, filled for ≤ and ≥.

Key takeaways

Solve inequalities like equations.
Flip the sign when ×/÷ by a negative.
Open circle = not included; filled = included.

Test yourself

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