Solving Inequalities

How to Solve Inequalities

In order to solve an inequality you first need to know what the symbols mean. The following descriptions are essential in being able to solve inequalities:

·         > means “greater than”,

·         \geq means “greater than or equal to”,

·         < means “less than”,

·         \leq means “less than or equal to”.

Inequalities are not always presented to us in a straight forward way. More often than not, they’re all jumbled up – like equations often are – and therefore they need to be rearranged and solved – also like equations.

When rearranging an inequality, you’re allowed to perform the same operation to both sides without changing it (just like an equation) with one exception: if you multiply/divide by a negative number, then the sign changes direction. In other words, < would become > and \leq would become \geq. For example, if you have

-3x \leq 6

and you divide both sides by -3, then the inequality sign flips and you get

x \geq -2

It is possible to avoid this scenario completely but sometimes it sneaks up on you, so this is important to know.

Example: Solve the inequality 5a - 4 > 2a + 8. Display your answer on a number line.

So, we solve it like we would an equation, Firstly, add 4 to both sides of the inequality to get

5a > 2a + 12

Then, subtract 2a from both sides to get

3a > 12

Finally, divide both sides by 3 to get

a > 4

When the question asks you to solve the inequality, your answer should be an inequality – a range of values that satisfy the inequality in the question. To see this answer working, if we pick a number which is greater than 4 (as the answer requires), say 5, and sub it into the left-hand side of the original inequality we get

5a - 4 = 5(5) - 4 = 25 - 4 = 21

Then, into the right-hand side:

2a + 8 = 2(5) + 8 = 10 + 8 = 18

Therefore, as the inequality in the question says, we get that

5a - 4 > 2a + 8.