Inequalities and number lines

Inequalities describe a range of answers, like 'at least 12' or 'under 18'. Here is how to show them on a number line and solve them.

Practise this lesson free (7 questions)

Showing inequalities

x < 3: x is less than 3. x ≥ −1: x is −1 or more.
On a number line, an open circle means the number is not included (< or >). A filled circle means it is included (≤ or ≥).
x ≥ −1: draw a filled circle at −1 (marked below), then an arrow to the right.
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Double inequalities

−2 ≤ x < 3 means x is between −2 and 3, including −2 but not 3.
The integers that satisfy it are −2, −1, 0, 1 and 2.

Solving

Solve like an equation: do the same to both sides.
3x − 4 ≥ 11 → 3x ≥ 15 → x ≥ 5.
If you multiply or divide both sides by a negative number, the inequality sign reverses. To avoid this, move the x term to the side where it will be positive:
10 − 2x > 4 → 10 > 4 + 2x → 6 > 2x → 3 > x, so x < 3.
Worked example

Solve 5 < 2x + 1 ≤ 13 and list the integer solutions.

  1. Subtract 1 from all three parts: 4 < 2x ≤ 12
  2. Divide all three parts by 2: 2 < x ≤ 6
  3. Integers greater than 2, up to and including 6

Answer: 2 < x ≤ 6, so x = 3, 4, 5, 6

Key idea

Open circle: not included. Filled circle: included. Solve like an equation, but the sign flips if you multiply or divide by a negative.

Key words
Range
The largest value minus the smallest value. It measures how spread out the data is.
Equation
A statement with an equals sign that is true only for some values of the letter, such as 2x + 1 = 9.
Negative number
A number less than zero, such as −4.
Inequality
A statement using <, >, ≤ or ≥ that compares two values, such as x > 3.
Term
One part of an expression or sequence. In 3x + 5, the terms are 3x and 5.

All KS3 Mathematics key terms and definitions →

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  1. A number line has an open circle at −1 and an arrow pointing right. Which inequality does it show?

    1. x < −1
    2. x ≥ −1
    3. x > −1
    4. x ≤ −1
    Show the answer

    Answer: x > −1

    Open circle: not included. Right: greater than.

  2. Solve 2x + 5 ≥ 13.

    1. x ≥ 4
    2. x ≥ 8
    3. x ≤ 4
    4. x ≥ 9
    Show the answer

    Answer: x ≥ 4

    2x ≥ 8, so x ≥ 4.

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Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.

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