IGCSEMathsEdexcelMaths AHigherRevision NotesEquations, Formulae & IdentitiesSolving InequalitiesSolving Quadratic Inequalities

Solving Quadratic Inequalities (Edexcel IGCSE Maths A: Higher): Revision Note

Q: How do you know whether to write the solution as one inequality or two separate inequalities?

Sketch the graph and shade the part you need: one connected piece between the intercepts means one inequality with x in the middle, two separate outer pieces mean two inequalities joined by "or". Using this page's example, with intercepts at 2 and 5 on a U-shaped curve: - (x − 2)(x − 5) ≤ 0 needs the part below the x-axis, which is the single middle section: 2 ≤ x ≤ 5 - (x − 2)(x − 5) ≥ 0 needs the part above the x-axis, which is the two outer sections: x ≤ 2 or x ≥ 5 Quick pattern for a U shape: - or ≥ : outside the intercepts, two parts joined by "or" For an ∩ shape it flips — 4 − x² > 0 is above the axis in the middle, giving −2 < x < 2. If the question uses strict signs, keep them strict in your answer.

Q: How do you find the x-intercepts of a quadratic graph?

Replace the inequality sign with an equals sign and solve — the solutions are the x-intercepts. 1. Rearrange so one side is 0 2. Solve = 0, usually by factorising: double brackets, the difference of two squares, or taking out a common factor of x 3. Set each bracket equal to 0 to get the x-values 4. If it will not factorise, use the quadratic formula Worked example from this page: 3x² + 2x − 6 > x² + 4x − 2 - Rearrange: 2x² − 2x − 4 > 0 - Divide by 2: x² − x − 2 > 0 - Solve x² − x − 2 = 0, so (x − 2)(x + 1) = 0 - The x-intercepts are x = 2 and x = −1 - It is a U shape and you want the parts above the axis, so x 2 Another page example: 4 − x² = 0 gives x² = 4, so x = ±2.

Exam code: 4MA1

Written by: Mark Curtis

Reviewed by: Dan Finlay

Updated on 27 January 2026

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Solving quadratic inequalities

What are quadratic inequalities?

A quadratic inequality has the form $a x^{2} + b x + c \geq 0$
- There is an $x^{2}$ term and any inequality sign, $\geq, \leq, >, <$
- They can usually be factorised
  - For example, $(x - 2) (x - 5) \geq 0$
Solutions to quadratic inequalities are ranges of $x$ values
- For example, $1 \leq x \leq 4$

How do I solve a quadratic inequality?

Quadratic inequalities are solved by sketching a graph
- The solutions then appears along the $x$ -axis
For example, to solve $(x - 2) (x - 5) \geq 0$
- Sketch the graph of $y = (x - 2) (x - 5)$
  - Show the $x$ -intercepts of 2 and 5
  - These come from solving $(x - 2) (x - 5) = 0$
- As the inequality is $\geq 0$ , shade any parts of the curve that are above the $x$ -axis
  - These are left of $x = 2$ and right of $x = 5$
- The solutions are the ranges on the $x$ -axis for those shaded parts
  - $x \leq 2$ or $x \geq 5$
  - Use the word "or" for two separate parts
  - In set notation this is ${x : x \leq 2} \cup {x : x \geq 5}$

Graph of quadratic inequality (x-2)(x-5) ≥ 0, showing shading for x ≤ 2 and x ≥ 5, and the quadratic function y = (x-2)(x-5) plotted, crossing x at 2 and 5.

What do I do if the sign of the inequality changes?

If the quadratic inequality is $a x^{2} + b x + c \geq 0$ where $a$ is positive
- then shade the parts of the curve above the $x$ -axis
If the quadratic inequality is $a x^{2} + b x + c \leq 0$ where $a$ is positive
- then shade the part of the curve below the $x$ -axis
For example, to solve $(x - 2) (x - 5) \leq 0$
- Repeat the process above but shade where the curve is below the $x$ -axis
  - The solution is $2 \leq x \leq 5$
  - (You do not use the word "or", but could say $x \geq 2$ "and" $x \leq 5$ )
  - In set notation this is ${x : x \geq 2} \cap {x : x \leq 5}$ , or alternatively ${x : 2 \leq x \leq 5}$

Graph depicting the quadratic inequality (x - 2)(x - 5) ≤ 0. The highlighted area shows y = (x - 2)(x - 5) between x = 2 and x = 5, inclusive.

If strict inequality signs are used, $>$ or $<$
- then you must use strict inequalities in your final answer

How can quadratic inequalities be made harder?

You may need to bring all the terms to one side first
- For example $x^{2} - x < 6$ becomes $x^{2} - x - 6 < 0$
  - It is easier to pick the side with a positive $x^{2}$
You may have to factorise the quadratic inequality first
- This may involve factorising
  - $a x^{2} + b x + c$ into double brackets
  - $a^{2} x^{2} - b^{2} = (a x + b) (a x - b)$ (the difference of two squares)
  - $a x^{2} + b x = x (a x + b)$ (factorising out an $x$ )
- If it does not factorise, you may need the quadratic formula to find $x$ -intercepts
You may be given a quadratic inequality $a x^{2} + b x + c \geq 0$ where $a$ is negative
- This means the sketch will be an $\cap$ shape
For example, solve $4 - x^{2} > 0$
- The $x$ intercepts are found from $4 - x^{2} = 0$ giving $x = \pm 2$
- The same process is then applied, but the graph has an $\cap$ shape

Graph of quadratic inequality 4 - x^2 > 0. The shaded region above y = 4 - x^2 shows valid x values between -2 and 2, excluding the endpoints.

Examiner Tips and Tricks

Think of inequalities with $x^{2}$ terms as graph-sketching questions in the exam.

Worked Example

Find the set of values for which $3 x^{2} + 2 x - 6 > x^{2} + 4 x - 2$

Answer:

Rearrange to get a zero on one side

$2 x^{2} - 2 x - 4 > 0$

Divide everything by 2 as it is a common factor

$x^{2} - x - 2 > 0$

Find the roots of the quadratic

$\begin{array}{l} x^{2} - x - 2 = 0 \\ (x - 2) (x + 1) = 0 \\ x = 2, x = - 1 \end{array}$

Sketch the graph and identify the parts which are above the x-axis

Graph of a red zigzag line in a U-shape, crossing the x-axis at -1 and 2, on a coordinate plane with arrows showing positive axes directions.

$x < - 1 or x > 2$

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Questions students ask about Solving Quadratic Inequalities

Real questions from students revising this topic on Save My Exams

How do you know if a quadratic graph is u-shaped or n-shaped?

Look at the number in front of x²: positive gives a U shape, negative gives an ∩ shape.

Write the inequality as ax² + bx + c with 0 on the other side
a positive (x², 3x²): U shape
a negative (−x², −2x²): ∩ shape

Examples from this page:

(x − 2)(x − 5) ≥ 0 expands to x² − 7x + 10, so the x² term is positive and the sketch is a U
4 − x² > 0 has −x², so the sketch is an ∩

Make life easier: when you rearrange, choose the side that gives a positive x², so you always sketch a U. For example, adding x² to both sides of 4 − x² > 0 gives 4 > x²; subtracting 4 from both sides then gives 0 > x² − 4, which is x² − 4 < 0 — the same answer, −2 < x < 2.

How do you know whether to write the solution as one inequality or two separate inequalities?

Sketch the graph and shade the part you need: one connected piece between the intercepts means one inequality with x in the middle, two separate outer pieces mean two inequalities joined by "or".

Using this page's example, with intercepts at 2 and 5 on a U-shaped curve:

(x − 2)(x − 5) ≤ 0 needs the part below the x-axis, which is the single middle section: 2 ≤ x ≤ 5
(x − 2)(x − 5) ≥ 0 needs the part above the x-axis, which is the two outer sections: x ≤ 2 or x ≥ 5

Quick pattern for a U shape:

< or ≤ : between the intercepts, x in the middle
> or ≥ : outside the intercepts, two parts joined by "or"

For an ∩ shape it flips — 4 − x² > 0 is above the axis in the middle, giving −2 < x < 2.

If the question uses strict signs, keep them strict in your answer.

How do you find the x-intercepts of a quadratic graph?

Replace the inequality sign with an equals sign and solve — the solutions are the x-intercepts.

Rearrange so one side is 0
Solve = 0, usually by factorising: double brackets, the difference of two squares, or taking out a common factor of x
Set each bracket equal to 0 to get the x-values
If it will not factorise, use the quadratic formula

Worked example from this page: 3x² + 2x − 6 > x² + 4x − 2

Rearrange: 2x² − 2x − 4 > 0
Divide by 2: x² − x − 2 > 0
Solve x² − x − 2 = 0, so (x − 2)(x + 1) = 0
The x-intercepts are x = 2 and x = −1
It is a U shape and you want the parts above the axis, so x < −1 or x > 2

Another page example: 4 − x² = 0 gives x² = 4, so x = ±2.