Inequalities

An inequality is like an equation, but instead of an equals sign, it uses an inequality symbol to show that one side is greater or smaller than the other. Inequalities are a core part of the AQA GCSE Mathematics specification and are tested on both tiers. Higher tier students also need to handle quadratic inequalities and represent solutions on graphs.

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Inequality Symbols

Symbol	Meaning	Example
<	Less than	x < 5
>	Greater than	x > 3
<=	Less than or equal to	x <= 7
>=	Greater than or equal to	x >= -2

Key Difference from Equations

An equation such as x + 3 = 7 has one solution (x = 4). An inequality such as x + 3 < 7 has a range of solutions (x < 4 — meaning any value less than 4).

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Representing Inequalities on a Number Line

When drawing an inequality on a number line:

• Use an open circle (o) for < or > (the value is NOT included).
• Use a closed circle (filled in) for <= or >= (the value IS included).
• Draw an arrow showing the direction of all the values that satisfy the inequality.

Inequality	Number Line Representation
x > 3	Open circle at 3, arrow pointing right
x <= 5	Closed circle at 5, arrow pointing left
-1 < x <= 4	Open circle at -1, closed circle at 4, line between them

Exam Tip: The AQA mark scheme is very strict about open and closed circles on number lines. An open circle when it should be closed (or vice versa) will cost you a mark. Always double-check the symbol.

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Solving Linear Inequalities

Solving an inequality follows the same rules as solving an equation, with one crucial exception: if you multiply or divide by a negative number, you must reverse the inequality sign.

Worked Example 1

Solve: 2x + 3 < 11

Subtract 3 from both sides: 2x < 8

Divide by 2: x < 4

Answer: x < 4

Worked Example 2

Solve: 5x - 7 >= 13

Add 7: 5x >= 20

Divide by 5: x >= 4

Answer: x >= 4

Worked Example 3

Solve: 3 - 2x > 7

Subtract 3: -2x > 4

Divide by -2 and reverse the sign: x < -2

Answer: x < -2

Exam Tip: Dividing or multiplying by a negative number reverses the inequality sign. This is the single most common mistake in inequality questions. If you are unsure, substitute a value from your answer to check it satisfies the original inequality.

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Solving Double Inequalities

A double inequality has the unknown in the middle, like -3 < 2x + 1 <= 7. Solve by performing the same operations on all three parts.

Worked Example 4

Solve: -3 < 2x + 1 <= 7

Subtract 1 from all parts: -4 < 2x <= 6

Divide all parts by 2: -2 < x <= 3

Answer: -2 < x <= 3

This means x is greater than -2 but less than or equal to 3.

Worked Example 5

Solve: 1 <= 3x - 2 < 13

Add 2: 3 <= 3x < 15

Divide by 3: 1 <= x < 5

Answer: 1 <= x < 5

Finding Integer Solutions

If the question asks for integer solutions, list all the whole numbers that satisfy the inequality.

For -2 < x <= 3 where x is an integer: x = -1, 0, 1, 2, 3

Note: -2 is NOT included (strict inequality <) but 3 IS included (<=).

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Inequalities and Regions on Graphs [H]

On the Higher tier, you may be asked to shade a region on a coordinate grid that satisfies one or more inequalities.

Rules for Graphing Inequalities

Inequality type	Line style	Shade
y < mx + c	Dashed line (not included)	Below the line
y > mx + c	Dashed line (not included)	Above the line
y <= mx + c	Solid line (included)	Below the line
y >= mx + c	Solid line (included)	Above the line

Worked Example 6

Show the region that satisfies all of: y >= 1, x < 4, and y < x + 2.

Step 1: Draw y = 1 as a solid horizontal line; shade above.

Step 2: Draw x = 4 as a dashed vertical line; shade to the left.

Step 3: Draw y = x + 2 as a dashed line; shade below.

The required region is the overlap of all three shaded areas.

flowchart TD
    A[Draw each boundary line] --> B{Solid or dashed?}
    B -- "= included (solid)" --> C[Solid line]
    B -- "not included (dashed)" --> D[Dashed line]
    C --> E[Shade correct side]
    D --> E
    E --> F[Identify the overlapping region]
    F --> G[Label the region R]

Exam Tip: AQA often asks you to shade the region that satisfies the inequalities and label it R. If multiple regions need shading, read the question carefully — sometimes you shade the unwanted region and leave the valid region unshaded.

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Quadratic Inequalities [H]

To solve a quadratic inequality, first solve the corresponding quadratic equation to find the critical values, then determine which intervals satisfy the inequality.

Worked Example 7

Solve: x^2 - 5x + 6 < 0

Step 1: Solve x^2 - 5x + 6 = 0

(x - 2)(x - 3) = 0

x = 2 or x = 3

Step 2: The quadratic x^2 - 5x + 6 is a U-shaped parabola. It is negative (below the x-axis) between the roots.

Answer: 2 < x < 3

Worked Example 8