## Key points

Inequalities are used to describe the relationship between expressions that are not equal. They can be presented algebraically, in words, or on a number line.

The two sides in an are equal. This is not the case in an inequality, as one side may be greater than (or less than) the other.

An inequality can have multiple solutions. When a single inequality is used there can be an number of solutions. In some cases, the number of solutions can be . For example, when the unknown value is an and between two given values.

Solutions to an inequality may be given as a simplified inequality, a finite list of values, a or as an illustration on a number line.

Back to top

## Video

Watch the video to learn how Shahan, a video games developer, uses inequalities in his work and find out why inequalities are important when it comes to designing new video games.

Back to top

## Understanding and using inequality symbols

Inequality symbols are used when values are not equal.

The used when writing inequalities is: ≠, <, ≤, > or ≥, where:

**≠**means ‘not equal to’**<**means ‘less than’**>**means ‘greater than’**≤**means ‘less than or equal to’**≥**means ‘greater than or equal to’

### Examples

End of image gallery

Which inequality is used to state that 𝒏 is at least 8?

Back to top

## Inequalities on a number line and listing integer solutions

Inequalities can be illustrated on a .

- For a single inequality, with one value:

Draw a circle above the value on the number line.

- The circle is unshaded for < and >
- The circle is shaded for ≤ and ≥

Draw a horizontal line with an arrow in the same direction as the inequality:

- to the left with an arrow for < and ≤
- to the right with an arrow to the right for > and ≥

For a single inequality there are an infinite number of integer solutions. These can’t be listed in a solution set.

- For an inequality with two values:

Draw circles above the values on the number line.

- The circle is unshaded for < and >
- The circle is shaded for ≤ and ≥

Draw a horizontal line between the circles.

For an inequality with two values there are a finite number of integer solutions. This can be written as a solution set which is presented in curly brackets. For example, {2, 3, 4}.

### Examples

End of image gallery

### Question

What inequality is shown on the number line?

Back to top

## Solving simple linear inequalities

A **single inequality** written with an may be solved by using :

Undo each process in the inequality by using the inverse operation. Inverse operations include addition and subtraction, and multiplication and division.

Start with the last operation and work back to the first.

For each step make sure that the inverse is applied to both sides of the inequality.

Once the variable is write the solution as a simple inequality.

An **expression between inequalities** is solved using inverse operations in the same way:

- Apply each inverse operation to all three parts of the inequality.
- Write the solution as a variable between two inequalities.

### Examples

End of image gallery

### Question

Solve the inequality and list the integer solutions for 𝒏

Back to top

## Practise using inequalities

### Quiz

Back to top

### Game - Divided Islands

Divided Islands. gameDivided Islands

Use your maths skills to help the islanders of Ichi build bridges and bring light back to the islands in this free game from BBC Bitesize.

Back to top