GCSE — mathsFoundation2 min read

Solving Inequalities

Solve linear and quadratic inequalities for GCSE Maths. Represent solutions on number lines.

Tutor AI Team2026-02-25T01:44:39.640Z

Inequalities are like equations but use $<$ , $>$ , $\leq$ , $\geq$ instead of $=$ . Solving inequalities follows the same steps as equations with one crucial difference: multiplying or dividing by a negative number flips the inequality sign.

Core Concepts

Linear Inequalities

Solve like equations:

$3x + 4 > 13$ → $3x > 9$ → $x > 3$

$5 - 2x \leq 11$ → $-2x \leq 6$ → $x \geq -3$ (sign flips!)

Double Inequalities

$4 < 2x + 1 \leq 9$

Subtract 1: $3 < 2x \leq 8$

Divide by 2: $1.5 < x \leq 4$

Number Line Representation

Open circle ○ for $<$ or $>$ (not including the value).
Closed circle ● for $\leq$ or $\geq$ (including the value).

Quadratic Inequalities (Higher)

$x^2 - 4x - 5 < 0$

Factorise: $(x - 5)(x + 1) < 0$

Roots: $x = 5$ and $x = -1$ .

The quadratic is negative between the roots: $-1 < x < 5$ .

For $> 0$ : the solution is outside the roots: $x < -1$ or $x > 5$ .

Worked Example: Example 1

Problem

$4x - 7 \geq 5$ → $4x \geq 12$ → $x \geq 3$

Solution

Worked Example: Example 2

Problem

$x^2 - 9 > 0$ → $(x-3)(x+3) > 0$ → $x < -3$ or $x > 3$

Solution

Practice Problems

1. Solve $2x + 3 < 11$ .
1. Solve $-3x \geq 12$ .
1. Solve $x^2 - x - 6 \leq 0$ .
1. Solve $1 \leq 3x - 2 < 10$ .

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Key Takeaways

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Solve like equations, but flip the sign when multiplying/dividing by a negative.
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Number lines: open circle = not included, closed circle = included.
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Quadratic inequalities: factorise, find roots, then determine the sign between/outside roots.

Tags:

#gcse#maths#algebra#inequalities

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