# Inequalities

## Topic Overview

An **inequality** is the relation between two values when they are not equal. There are various mathematical symbols used to describe inequalities:

- The symbol < means that the number to the left of the symbol is '
**less than**' the number to the right of the symbol. For example:

2 < 4 means that 2 is less than 4.

- The symbol ≤ means that the number to the left of the symbol is 'less than or equal to' the number to the right of the symbol. For example:

2 ≤ 4 and 2 ≤ 2 are both mathematically correct because 2 is less than 4 and also equal to 2.

- The symbol > means that the number to the left of the symbol is 'greater than' the number to the right of the symbol. For example:

5 > 3 means that 5 is greater than 3.

- The symbol ≥ means that the number to the left of the symbol is 'greater than or equal to' the number to the right of the symbol. For example:

5 ≥ 3 and 3 ≥ 3 are both mathematically correct because 5 is greater than 3 and also 3 is equal to 3.

Properties

Inequalities have a variety of different properties, which are listed below:

- When you link two different inequalities which are in order, you can group them together. For example:

**a > b and b > c can be grouped together to become a > c**

When you reverse inequalities, you need to make sure that the symbol is still pointing at the correct value. For example:

**If you reverse a > b then b < a**

- According to the 'Law of Trichotomy', an inequality must be one, and ONLY one, of the following expressions:

**a < b OR a = b OR a > b**

- When adding and subtracting inequalities, you must adhere to the following rules:

**If a < b, then a - c < b - c**

**If a > b, then a + c > b + c**

**If a > b, then a - c > b - c**

- Similarly, when multiplying and dividing inequalities, you must adhere to the following rules:

**If a < b, and c is positive, then a x c < b x c**

**If a < b, and c is negative, then a x c > b x c**

- When you begin to introduce negative values into inequalities, this process changes the direction of the inequality. For example:

**If a < b then -a > -b**

**If a > b then -a < -b**

Similarly, when you take the reciprocal of both a and b, this can change the direction of the inequality. For example:

**If a < b, then 1/a > 1/b**

**If a > b, then 1/a < 1/b**

However when either a or b is negative (but not both) the direction stays the same. For example:

**If a < b, then 1/a < 1/b**

**If a > b, then 1/a > 1/b**

- A square of a number will be greater than or equal to zero. For example:

**a^2 ≥ 0**

- Taking a square root will not change the inequality (but only when both a and b are greater than or equal to zero). For example:

**If a ≤ b then √a ≤ √b**

(for a,b ≥ 0)

## Key Concepts

In the new linear GCSE Maths paper, you will be required to solve various mathematical problems involving inequalities. The specific questions you will be expected to answer will vary depending upon which examination board with which you are registered, but as a rule you will be required to:

- Solve linear inequalities in one variable, and represent the solution set on a number line
- Solve linear inequalities in two variables, and represent the solution set on a co-ordinate grid

Listed below are a series of summaries and worked examples to help you solidify your knowledge about inequalities.