 ## What you need to know

Inequalities on a Number Line & Solving Inequalities

Whilst an equation is a mathematical statement that involves an equals sign, an inequality is one that involves one of the following 4 symbols: $<, \leq, >, \geq$. 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 a 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.

Solving Inequalities

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 $\leq$ 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 $5a - 4 > 2a + 8$.

## 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. 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.

We solve it as we would an equation. Firstly, add $5k$ to both sides to get

$7 + 2k > 12$

Then, subtract 7 from both sides to get

$2k > 5$

Finally, divide both sides by 2 to get

$k > \dfrac{5}{2} (= 2.5)$

As $k$ must be bigger than 2.5, the smallest whole number that satisfies this inequality is 3.

## Inequalities On a Number Line and Solving Inequalities Teaching Resources

The inequality number line questions and solving inequality worksheets can be utilised by classroom teachers and personal tutors. Regardless of whether you are a Maths tutor in York or a GCSE teacher in London, these revision questions could be incorporated into your resource bank to be used as homework and starters at the beginning of a lesson.