Inequalities

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 ≥ 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 .

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