Whilst an equation is a mathematical statement that involves an equals sign, an inequality is one that involves one of the following 4 symbols: . Their definitions:
- means “greater than”,
- means “greater than or equal to”,
- means “less than”,
- means “less than or equal to”.
For example, the inequality means “2 is less than 5”, which is obviously true. This isn’t all that interesting, really. The more interesting inequalities you’ll see are like
which means that “ is less than or equal to 8”. So, might be 7 or -10 or 2.34 or or… the list would go on forever, literally. It is called a range of values and is best expressed by (to save us writing forever).
Note: the inequality includes the value 8 as a possibility for , whilst the inequality does not. We call and inclusive inequalities, and we call and strict inequalities.
A common way to display an inequality is using a number line. On a number line, looks like
The circle tells us where the end of the inequality is. For an inclusive inequality, use a closed circle like ●, and for a strict inequality, use an open circle like ○. Then, the arrow is used to explain whether to include all the numbers below or all the numbers above where the circle is placed – an arrow to the left includes all the numbers below, and an arrow to the right includes all the numbers above.
Example: Display the inequality on a number line.
This inequality looks like two inequalities put together – and that’s exactly what it is. The first part is , which means that “3 is less than or equal to ” – if you think about what’s being said here (or alternatively you could turn the inequality around), then this is the same thing as saying . Then, the second part is , which means “ is less than 10″.
So, we must have being bigger than or equal to 3 and being less than 10, i.e. it must be somewhere in between (including 3 itself). On a number line, this looks like
One part of the inequality was inclusive and the other was strict, so we have one open circle and one closed circle. As you can see, there is no need for an arrow in this situation, since the range of values is enclosed between 3 and 10 and does not get any bigger or smaller.
Inequalities are not always presented to us in the way they are above. More often than not, they’re all jumbled up – like equations often are – and 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 would become . For example, if you have
and you divide both sides by -3, then the inequality sign flips and you get
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 . 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
Then, subtract from both sides to get
Finally, divide both sides by 3 to get
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
Then, into the right-hand side:
Therefore, as the inequality in the question says, we get that .