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Maths I

Simple equations and inequalities

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# Solving inequalities

## Introduction

With some inequalities, we deal with one number being less than another number, or with one number being greater than another number.

• < stands for 'is less than'

• > stands for 'is greater than'

For example, and

For the inequality , can be any number less than .

## Example 1

Solve the inequality .

times x is less than .

Divide both sides by .

is less than .

So the solution is:

.

So is any number less than .

## Example 2

Solve the inequality .

We want the term on the left-hand side, by itself.

So, take away from the left-hand side.

To balance the equation, we must do the same to the right hand side.

So, take away from the right-hand side.

This gives us,

So, .

The solution is .

## Example 3

Solve the inequality .

Add to both sides, to leave on the left, by itself:

So, .

Therefore, the solution is .

## Example 4

Solve the inequality .

Either:

(multiplying out the brackets.)

So, (subtracting from both sides.)

Therefore, .

Or:

Divide both sides of by , to get

.

Therefore

### Now, try to solve these inequalities.

Question

Solve .

Answer

Question

Solve

Answer

(because .)

Question

Solve.

Answer

(because .)

Question

Solve .

Answer

(either from and , or from )

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