Inequalities On a Number Line

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What you need to know

Inequalities on a Number Line

Before learning about how to display inequalities on a number line, you firstly need to know the following symbols and their definitions :

·         $>$ means “greater than”,

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

·         $<$ means “less than”,

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

For example, the inequality $2<5$ 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

$x\leq 8$,

which means that “$x$ is less than or equal to 8”. So, $x$ might be 7 or -10 or 2.34 or $\pi$ or… the list would go on forever, literally. It is called a range of values and is best expressed by $x \leq 8$ (to save us writing forever).

Note: the inequality $x\leq 8$ includes the value 8 as a possibility for $x$, whilst the inequality $x<8$ does not. We call $\leq$ and $\geq$ inclusive inequalities, and we call $<$ and $>$ strict inequalities.

A common way to display an inequality is using a number line. On an inequality number line, $x\leq 8$ looks like:

The circle tells us where the end of the inequality is. For an inclusive inequality, use a closed circle like $\bullet$, and for a strict inequality, use an open circle like $\circ$. 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 $3 \leq x < 10$ on a number line.

This inequality looks like two inequalities put together – and that’s exactly what it is. The first part is $3\leq x$, which means that “3 is less than or equal to $x$” – 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 $x \geq 3$. Then, the second part is $x<10$, which means “$x$ is less than 10”.

So, we must have $x$ 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.

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Example Questions

The first inequality, $y>3$, will require an open circle at 3 and an arrow pointing right.

The other inequality, $y<-2$, will require an open circle at -2 and an arrow pointing left.

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We solve it as we would an equation. Firstly, subtract $x$ from both sides to get

$12 \geq 5x - 18$

Then, add 18 to both sides to get

$30 \geq 5x$

Finally, divide both sides by 5 to get

$6 \geq x$

This can also be expressed like $x \leq 6$, so to display this on a number line we will need a closed circle at 6 and an arrow pointing to the left.

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