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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Inequalities

Inequalities

The basics of inequalities:

• - means a is less than b (so b is greater than a)

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

• - means a is greater than or equal to b etc

• - means a is greater than b etc

Solving Inequalities

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. This is all done the same way as with equations. There is one difference however - if you multiply or divide by a negative number, the inequality sign is reversed.

Example

Solve: .

(note: sign reversed because we divided by -2)

Ranges of Values

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

,

means is greater than or equal to 4 but less than 10.

If was limited to whole numbers, then it could be either 4, 5, 6, 7, 8 or 9, but not 10.

Graphs of Inequalities

Inequalities are represented on graphs using shading. For example, if , the graph of would be drawn. Then we shade a region of the graph depending on what we are asked to do. If we need to shade the points the inequality represents, then we'd shade the points here is larger than (i.e. above the line). If we shade the region not represented by the inequality, we'd shade the area where is less than (i.e. the region below the line).

Example

and

(NB: this is the same as the two inequalities and )

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).

(diagram of this graph is needed here)

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.

This can be split into the two inequalities:

and

And so:

and

therefore:

and .

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.

(diagram needed illustrating this example.)

The solution to the pair of inequalities occurs where the two lines overlap on the number line, i.e. for .

Comments

This article is rather jumbled in terms of order. It needs structuring to have the easier parts on ranges of values and number lines first and graphs and solving inequalities later on. It also needs a greater range of examples everywhere and a section talking about practical uses of inequalities (with examples) as well as mentioning using dashed and solid lines when drawing the graphs. Finally, it needs graphs making for some of the current examples.