a < b means a is less than b (so b is greater than a)

a £ b means a is less than or equal to b (so b is greater than or equal to a)

a ³ b means a is greater than or equal to b etc.

a > b means a is greater than b etc.

If you have an inequality, you can add or subtract numbers from each side of the inequality, as with an equation. You can also multiply or divide by a constant. However, if you multiply or divide by a negative number, the inequality sign is reversed.

*Example*

Solve 3(x + 4) < 5x + 9

3x + 12 < 5x + 9

\ -2x < -3

\ __x > 3/2__ (note: sign reversed because we divided by -2)

Inequalities can be used to describe what range of values a variable can be.

E.g. 4 £ x < 10, means x is greater than or equal to 4 but less than 10.

**Graphs**

Inequalities are represented on graphs using shading. For example, if y > 4x, the graph of y = 4x would be drawn. Then either all of the points greater than 4x would be shaded or all of the points less than or equal to 4x would be shaded.

*Example*:

x + y < 7

and 1 < x < 4 (NB: this is the same as the two inequalities 1 < x and x < 4)

Represent these inequalities on a graph by leaving unshaded the required regions (i.e. do not shade the points which satisfy the inequalities, but shade everywhere else).

**Number Lines**

Inequalities can also be represented on number lines. Draw a number line and above the line draw a line for each inequality, over the numbers for which it is true. At the end of these lines, draw a circle. The circle should be filled in if the inequality can equal that number and left unfilled if it cannot.

*Example*:

On the number line below show the solution to these inequalities.

-7 £ 2x – 3 < 3

This can be split into the two inequalities:

-7 £ 2x – 3 and 2x – 3 < 3

\ -4 £ 2x and 2x < 6

\ -2 £ x and x < 3

The circle is filled in at –2 because the first inequality specifies that x can equal –2, whereas x is less than (and not equal to) 3 and so the circle is not filled in at 3.

The solution to the inequalities occurs where the two lines overlap, i.e. for -2 £ x < 3