Inequalities | CIE IGCSE Maths: Core Revision Notes 2025

Interpreting Inequalities

An inequality tells you that something is greater than (>) or less than (<) something else
- x > 5 means x is greater than 5
  - x could be 6, 7, 8, 9, ...
Inequalities may also include being equal (=)
- ⩾ means greater than or equal to
- ⩽ means less than or equal to
  - x ⩽ 10 means x is less than or equal to 10
    - x could be 10, 9, 8, 7, 6, ...
When they cannot be equal, they are called strict inequalities
- > and < are strict inequalities
  - x > 5 does not include 5 (strict)
  - x ⩾ 5 does include 5 (not strict)

You may be given two end points and have to list the integer (whole number) values of x that satisfy the inequality
Look at whether each end point is included or not
- 3 ⩽ x ⩽ 6
  - x = 3, 4, 5, 6
- 3 ⩽ x < 6
  - x = 3, 4, 5
- 3 < x ⩽ 6
  - x = 4, 5, 6
- 3 < x < 6
  - x = 4, 5
If only one end point is given, there are an infinite number of integers
- x > 2
  - x = 3, 4, 5, 6, ...
- x ⩽ 2
  - x = 2, 1, 0, -1, -2, ...
  - Remember zero and negative whole numbers are integers
  - If the question had said positive integers only then just list x = 2, 1
You may be asked to find integers that satisfy two inequalities
- 0 < x < 5 and x ⩾ 3
  - List separately: x = 1, 2, 3, 4 and x = 3, 4, 5, 6, ...
  - Find the values that appear in both lists: x = 3, 4
If the question does not say x is an integer, do not assume x is an integer!
- x > 3 actually means any value greater than 3
  - 3.1 is possible
  - π = 3.14159... is possible
You may be asked to find the smallest or largest integer
- The smallest integer that satisfies x > 6.5 is 7

List all the integer values of $x$ that satisfy

$- 4 \leq x < 2$

Integer values are whole numbers
-4 ≤ x shows that x includes -4, so this is the first integer

x = -4

x < 2 shows that x does not include 2
Therefore the last integer is x = 1

x = 1

For the answer, list all the integers from -4 to 1
Remember integers can be zero and negative

$x = - 4, - 3, - 2, - 1, 0, 1$

A number line representing an inequality

Draw circles above the end points and connect them with a horizontal line
- Leave an open circle for end points with strict inequalities, < or >
  - These end points are not included
- Fill in a solid circle for end points with ≤ or ≥ inequalities
  - These end points are included
Use a horizontal arrow for inequalities with one end point
- x > 5 is an open circle at 5 with a horizontal arrow pointing to the right

Represent the following inequalities on a number line.

(a)

- 2 \leq x < 1

-2 is included so use a closed circle

1 is not included so use an open circle

Number line from -2 to 1, not including -2

(b)

t < 3

3 is not included so use an open circle
There is no second end point
Any value less than three is accepted, so draw a horizontal arrow to the left