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### Inequalities

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 un-shaded 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 .

Copyright © Matthew Pinkney 2003

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