Solving InequalitiesEdexcel IGCSE Maths: Revision notes

Section 1

What is a linear inequality and how do you solve one?

A linear inequality compares two expressions using $<$ , $>$ , $\leq$ or $\geq$ instead of $=$ . You solve it exactly like an equation: do the same operation to both sides to isolate the variable.

Example: Solve $3x + 5 < 20$

$3x < 15$

$x < 5$

The solution is a range of values, not a single number. Any value of $x$ less than 5 satisfies the original inequality.

Key termsinequalitysolution set

Example

Solve $2x - 7 \geq 9$ . Add 7: $2x \geq 16$ . Divide by 2: $x \geq 8$ .

Section 2

Why does the inequality sign flip when multiplying or dividing by a negative number?

This is the single most important rule in this topic. When you multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign must reverse.

Example: Solve $-4x < 12$

Divide both sides by $-4$ and flip the sign: $x > -3$

Why? Consider $2 < 5$ . Multiply both sides by $-1$ : naively you'd get $-2 < -5$ , which is false. The correct statement is $-2 > -5$ . The sign must flip to keep the statement true.

Key termssign flip

Common mistake

Forgetting to flip the sign when dividing by a negative is the #1 exam error in this topic. Always check: did I multiply or divide by a negative? If yes, flip.

Exam tip

To avoid flipping altogether, rearrange so the $x$ term stays positive. E.g. instead of $-4x < 12$ , solve $12 > 4x$ then divide by positive 4.

Section 3

How do you show a solution set on a number line?

A number line represents an inequality using a circle at the boundary value and a line/arrow showing the range.

Open circle (unfilled) at the boundary means the value is not included — used for $<$ and $>$
Closed circle (filled/shaded) at the boundary means the value is included — used for $\leq$ and $\geq$
An arrow extends from the circle in the direction of all included values

Example: $x \geq -2$ is shown as a filled circle at $-2$ with an arrow pointing right (towards larger numbers).

Example: $x < 3$ is shown as an open circle at $3$ with an arrow pointing left (towards smaller numbers).

Key termsopen circleclosed circle

Think of it like this

Think of an open circle like a door that's shut — you can get right up to it but not through. A closed circle is a door propped open — you can stand right on the threshold.

Section 4

How do you solve a double (compound) inequality?

A double inequality like $-3 \leq 2x + 1 < 9$ has two conditions at once. Solve by applying the same operation to all three parts simultaneously.

Example: Solve $-3 \leq 2x + 1 < 9$

Subtract 1 from all three parts: $-4 \leq 2x < 8$

Divide all three parts by 2: $-2 \leq x < 4$

The solution set is all values of $x$ from $-2$ (inclusive) up to but not including $4$ .

Key termsdouble inequalitycompound inequality

Exam tip

Whatever you do to one part, do to all three parts — this keeps the inequality balanced, exactly like solving a normal equation.

Section 5

How do you solve a quadratic inequality?

For a quadratic inequality like $x^2 - 5x + 6 > 0$ :

Rearrange so one side is 0
Factorise: $(x-2)(x-3) > 0$
Find the critical values (roots): $x = 2$ and $x = 3$
Sketch the parabola (a $u$ -shape since the $x^2$ coefficient is positive) and identify which regions satisfy the inequality

Since the parabola is above the $x$ -axis (positive) outside the roots, the solution is $x < 2$ or $x > 3$ .

Example: Solve $x^2 - 5x + 6 < 0$ (same factorisation, opposite inequality). The parabola is below the axis (negative) between the roots, so the solution is $2 < x < 3$ .

Rule of thumb for a positive $x^2$ parabola: ' $> 0$ ' gives outside the roots; ' $< 0$ ' gives between the roots.

Key termsquadratic inequalitycritical values

Common mistake

Do not simply solve $(x-2)(x-3) > 0$ by saying ' $x - 2 > 0$ and $x - 3 > 0$ ' only — this misses the second valid region. Always sketch the parabola to check both regions.

Example

Solve $x^2 \geq 9$ . Rearrange: $x^2 - 9 \geq 0$ , factorise: $(x-3)(x+3) \geq 0$ , roots at $x = -3, 3$ . Outside the roots (u-shape, $\geq 0$ ): $x \leq -3$ or $x \geq 3$ .

Must Know

Solve linear inequalities the same way as equations, isolating the variable on one side
Flip the inequality sign whenever you multiply or divide both sides by a negative number
Open circle = value excluded ( $<, >$ ); closed circle = value included ( $\leq, \geq$ )
For double inequalities, apply the same operation to all three parts
To solve a quadratic inequality: rearrange to $= 0$ , factorise, find critical values, then sketch the parabola to find the correct region(s)
For a positive $x^2$ parabola: ' $>0$ ' means outside the roots, ' $<0$ ' means between the roots

That's the notes covered.

Carry on to the next subtopic.