Inequalities Explained: KS3 Maths Guide with Worked Examples

An inequality is a mathematical statement that compares two expressions using a symbol that shows one is greater than, less than, or equal to the other. Unlike an equation, which has one solution, an inequality has a whole range of solutions. Understanding inequalities is a key part of KS3 algebra, taught in Year 8 and Year 9.

What are the inequality symbols?

There are four inequality symbols used at KS3:

Symbol | Meaning | Example | Reads as
> | strictly greater than | x > 5 | x is greater than 5
< | strictly less than | x < 3 | x is less than 3
≥ | greater than or equal to | x ≥ 2 | x is at least 2
≤ | less than or equal to | x ≤ 7 | x is at most 7

Memory tip: the open end of the symbol points towards the larger value. So in 3 < 8, the symbol opens to the right towards 8 (the bigger number), and closes towards 3 (the smaller number). An easy way to remember: the crocodile eats the bigger number and the mouth always opens towards it.

How to represent inequalities on a number line

Number lines are the standard way to show the solution set of an inequality at KS3.

Rules for circles on number lines:

An open circle (○) means the endpoint is not included. Use it for > and <.

A filled circle (●) means the endpoint is included. Use it for ≥ and ≤.

Examples of number line representations

x > 3: open circle at 3, arrow pointing right (all values greater than 3, but not 3 itself).

x ≤ 5: filled circle at 5, arrow pointing left (all values up to and including 5).

1 ≤ x < 6: filled circle at 1, open circle at 6, shaded region between them (x is at least 1 and strictly less than 6).

Solving linear inequalities

You solve a linear inequality almost identically to a linear equation — the critical exception is when you multiply or divide by a negative number, which reverses the inequality sign.

The golden rule

Performing the same operation on both sides keeps the inequality balanced — unless that operation is multiplying or dividing by a negative, in which case the inequality symbol flips.

Worked example 1 — simple inequality

Solve x + 4 > 9.

Subtract 4 from both sides: x > 5

Solution set: all real numbers greater than 5. On a number line: open circle at 5, arrow pointing right.

Worked example 2 — inequality with multiplication

Solve 3x ≤ 18.

Divide both sides by 3 (positive, so sign stays the same): x ≤ 6

Solution set: all numbers less than or equal to 6.

Worked example 3 — two-step inequality

Solve 2x − 3 < 11.

Step 1 — Add 3 to both sides: 2x < 14

Step 2 — Divide by 2: x < 7

Solution set: all numbers strictly less than 7.

Worked example 4 — dividing by a negative number

Solve −4x > 20.

Divide both sides by −4. Because we divide by a negative, flip the inequality sign: x < −5

Check: Try x = −6. Then −4 × (−6) = 24 > 20 ✓. Try x = −4. Then −4 × (−4) = 16, and 16 > 20 is FALSE ✓ (correctly excluded).

Worked example 5 — two-step inequality with a negative coefficient

Solve 7 − 2x ≥ 1.

Step 1 — Subtract 7 from both sides: −2x ≥ −6

Step 2 — Divide by −2 (flip the sign!): x ≤ 3

Check: x = 3: 7 − 6 = 1 ≥ 1 ✓. x = 4: 7 − 8 = −1, and −1 ≥ 1 is FALSE ✓.

Worked example 6 — double inequality

Find all integers n such that −1 < 2n + 1 ≤ 9.

Step 1 — Subtract 1 from all three parts: −2 < 2n ≤ 8

Step 2 — Divide all three parts by 2: −1 < n ≤ 4

Integers in the solution: n = 0, 1, 2, 3, 4. (Note: n = −1 is excluded because the left-hand inequality is strict >.)

Integer solutions

At KS3, examiners often ask you to "list the integer values" that satisfy an inequality. An integer is a whole number (including negatives and zero). Always check whether the endpoint itself is included or excluded.

Inequality | Integer solutions (for small values)
x > 2 | 3, 4, 5, 6, ...
x ≥ 2 | 2, 3, 4, 5, ...
x < 2 | ..., −1, 0, 1
x ≤ 2 | ..., 0, 1, 2
−2 < x ≤ 3 | −1, 0, 1, 2, 3

How inequalities fit the KS3 national curriculum

The Department for Education's KS3 mathematics programme of study requires pupils to "use algebraic methods to solve linear equations in one variable" and "understand and use inequality notation." The statutory national curriculum for Key Stage 3 and 4 covers inequality symbols, solving, and number line representation as core algebra topics. Inequalities re-appear throughout GCSE algebra, including quadratic inequalities, so a secure foundation at KS3 pays dividends.

Common mistakes

Mistake 1 — Forgetting to flip the sign when dividing by a negative. This is the number one error with inequalities. Write a reminder next to every step that involves a negative divisor or multiplier.

Mistake 2 — Using an open circle when a closed circle is needed (or vice versa). x ≥ 4 uses a filled circle; x > 4 uses an open circle. The = in the symbol is the indicator.

Mistake 3 — Misreading ≤ as < or ≥ as >. Read the symbol carefully. A mark at KS3 and GCSE is often lost by listing x ≤ 5 as solutions 4, 3, 2, ... without including 5 itself.

Mistake 4 — Failing to check the answer. Pick one number inside your solution set and one outside and substitute both into the original inequality to confirm correctness.

Frequently asked questions

What is the difference between an equation and an inequality?

An equation uses an equals sign (=) and typically has one specific solution. For example, 2x = 8 has the single solution x = 4. An inequality uses >, <, ≥, or ≤ and has an infinite range of solutions (a solution set). For example, 2x > 8 is satisfied by x = 5, x = 6.3, x = 100, and infinitely many other values greater than 4.

When do I flip the inequality sign?

You flip the inequality sign only when you multiply or divide both sides by a negative number. Adding or subtracting any number (positive or negative) never changes the direction of the inequality. Multiplying or dividing by a positive number also leaves the direction unchanged. The flip rule is specific to multiplication and division by a negative.

What does it mean for x to satisfy an inequality?

A value of x satisfies an inequality if, when you substitute it in, the inequality statement is true. For x > 5, the value x = 7 satisfies it (because 7 > 5 is true), but x = 3 does not (because 3 > 5 is false). The full solution set is all values that make the statement true.

How do I find the integer solutions of a double inequality?

Solve the double inequality to get bounds on x, then list every whole number strictly between (or on) those bounds. For example, 1 < x ≤ 5 gives integers 2, 3, 4, 5. For −3 ≤ x < 2 the integers are −3, −2, −1, 0, 1 (2 is excluded because the right-hand inequality is strict). Always check each endpoint against the original symbols.

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