Linear Inequalities on a Number Line: KS3 Maths

An inequality shows that two expressions are not necessarily equal — one is greater than, less than, or equal to the other. A linear inequality in one variable (like 2x + 3 < 11) can be solved like a linear equation, with one crucial exception: dividing or multiplying by a negative number reverses the inequality sign.

What do the inequality symbols mean?

Symbol | Meaning | Example | In words
> | Strictly greater than | x > 4 | x is more than 4
< | Strictly less than | x < −2 | x is less than −2
≥ | Greater than or equal to | x ≥ 0 | x is at least 0
≤ | Less than or equal to | x ≤ 7 | x is at most 7

The difference between > and ≥ matters: x > 4 does not include 4 itself, while x ≥ 4 does.

How do you show an inequality on a number line?

A number line shows which values of x satisfy the inequality:

Open circle (○): the endpoint is NOT included. Use this for strict inequalities (> or <).

Closed circle (●): the endpoint IS included. Use this for ≥ or ≤.

Arrow or shaded line: extends in the direction of the valid values.

Examples:

x > 3: open circle at 3, arrow pointing right.

x ≤ −1: closed circle at −1, arrow pointing left.

−2 < x ≤ 5: open circle at −2, closed circle at 5, line between them.

How do you solve a one-step inequality?

Solve in the same way as a one-step equation: isolate x by performing the inverse operation.

Worked example 1 — solve x + 7 < 12:

x + 7 < 12 → x < 12 − 7 → x < 5

On the number line: open circle at 5, arrow pointing left.

Worked example 2 — solve 3x ≥ 18:

3x ≥ 18 → x ≥ 18 ÷ 3 → x ≥ 6

On the number line: closed circle at 6, arrow pointing right.

How do you solve a two-step inequality?

Worked example 3 — solve 2x − 5 > 7:

Add 5 to both sides: 2x > 12.

Divide both sides by 2: x > 6.

Answer: x > 6 — open circle at 6, arrow pointing right.

Worked example 4 — solve 3x + 4 ≤ 19:

Subtract 4: 3x ≤ 15.

Divide by 3: x ≤ 5.

Answer: x ≤ 5 — closed circle at 5, arrow pointing left.

What happens when you divide by a negative number?

This is the most important rule for inequalities. Dividing or multiplying both sides by a negative number reverses the inequality sign.

Worked example 5 — solve −2x < 8:

Divide both sides by −2 and FLIP the sign: x > −4.

Answer: x > −4 — open circle at −4, arrow pointing right.

To avoid this pitfall, you can alternatively add 2x to both sides and rearrange to avoid dividing by a negative: −2x < 8 → 0 < 8 + 2x → −8 < 2x → x > −4. Same answer, no sign flip needed.

How do you show a combined (double) inequality on a number line?

A combined inequality like −3 < x ≤ 4 means x is greater than −3 AND at most 4.

Mark an open circle at −3.

Mark a closed circle at 4.

Draw a line between the two circles.

The solution set contains all values strictly between −3 and 4, including 4 but not −3.

To solve a combined inequality, work on all three parts simultaneously:

Worked example 6 — solve −1 ≤ 2x + 3 < 9:

Subtract 3 from all three parts: −4 ≤ 2x < 6. Divide all three by 2: −2 ≤ x < 3.

On the number line: closed circle at −2, open circle at 3, line between them.

Frequently asked questions

Can the solution to an inequality be a single value?

If you solve an inequality and find, for example, x ≥ 7 and x ≤ 7, the only solution is x = 7. This is an edge case; most inequality questions produce a range of values. However, when two inequalities are combined and they share only one boundary value, that single value can be the answer.

How do you list the integer solutions of an inequality?

After solving, identify which whole numbers lie within the solution set. For −2 ≤ x < 5, the integer solutions are −2, −1, 0, 1, 2, 3, 4. Note that 5 is excluded (strict inequality) and −2 is included (≥). Listing integer solutions is a common exam question at KS3.

Why does multiplying by a negative flip the inequality?

Consider the true statement 2 < 6. Multiply both sides by −1: −2 and −6. On a number line, −2 is to the RIGHT of −6, meaning −2 > −6. The order reversed. This is because multiplying by a negative reflects all numbers across zero, swapping which side is larger.

What is the difference between an inequality and an equation?

An equation has a single value (or a finite set of values) as its solution: 2x + 3 = 11 gives exactly x = 4. An inequality gives a range of values: 2x + 3 < 11 gives x < 4, which is infinitely many values. Inequalities are represented by a region on a number line (or a region on a coordinate grid for two-variable inequalities).

For Socratic KS3 algebra practice including inequalities, see aitutors.me.