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 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 .
Copyright © Matthew Pinkney 2003
