A level revision

Inequalities

An inequality describes the relative size of two numbers or terms. Inequalities make use of the less than and more symbols. You need to know how to solve both linear and quadratic inequalities as well as double inequalities.

To solve an inequality you want to aim to make x the subject in the simplest form possible. You can do more or less what you would do when rearranging a normal expression but remember when dividing or multiplying by a negative number the inequality sign should flip.

Single linear inequalities

Single linear inequalities are the easier inequality to solve as the highest power you are dealing with is 1 and there is only one inequality.

Example question: Solve 2x + 8 < 4x + 10

Worked answer: To solve this inequality we must make x the subject:

2x + 8 < 4x + 10
subtract 8 from both sides: 2x < 4x + 2 subtract 4x from both sides: -2x < 2 divide by -2, remembering to flip the inequality sign: x > -1

Therefore the values which satisfy the original equality are all those where x > -1.

Double linear inequalities

Double linear inequalities involve two inequalities combined together such as 2x + 12 < 7x - 3 < 5x + 20. To solve double inequalities first split them into two single inequalities and solve each one on it's own:

First part: 2x + 12 < 7x - 3
Add 3 to both sides: 2x + 15 < 7x
Subtract 2x from both sides: 15 < 5x
Divide by 5: 3 < x

Second part: 7x - 3 < 5x + 20
Add 3 to both sides: 7x < 5x + 23
Subtract 5x from both sides: 2x < 23
Divide by 2: x < 11.5

We now have two inequalities: x > 3 and x < 11.5. You should now to combine the two back together again, it may not always be possible but in this case it is. x must be greater than 3 but also less than 11.5, therefore the final answer is: 3 < x < 11.5

Solving quadratic inequalities using graphs

To solve quadratic inequalities you need to first factorise. For example to solve x² + 3x - 10 >= 0 first factorise the quadratic to get (x + 5)(x - 2) >= 0. You should then sketch the graph of the equation. As a positive x² graph is must be in a U shape crossing the y axis at -10. From the factorised form it must cross the x axis at -5 and +2. You don't need to be terribly accurate with your drawing, just a sketch will do:

We want to find the values of x which satisfy x² + 3x - 10 >= 0. From the graph the x values where the graph is equal to or greater than 0 are at either side of where it crosses the x axis (orange lines) as well as the point at which the graph crosses the x axis too. Therefore the solution to the quadratic x =< -5 and x >= 2. These are two very separate inequalities which cannot be combined.

So to recap solving quadratic inequalities:

1. Factorise the quadratic
2. Draw a graph of the equation
3. Identify the areas which satisfy the inequality.
4. Write one or more inequalities to represent the areas, combing them if you can.