Solving Inequalities Worksheets | Questions and Revision | MME

# Solving Inequalities

Level 4 Level 5

## What you need to know

### 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. Understanding how to solve equations will help with this topic, and you also need to have the basic knowledge of how to display an inequality on a number line

### Take Note:

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

$x \geq -2$

### Example 1: Solving Inequalities

Solve the inequality $5a - 4 > 2a + 8$

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

$a > 4$

Note: When the question asks you to solve the inequality, your answer should be an inequality, avoid using an equals sign.

### Example 2: Solving Inequalities

Solve the inequality $3x^2 - 7 <20$

Firstly add 7 to both sides

$3x^2 <27$

The divide by 3

$x^2 <9$

Then finally square root

$x<3$

### Example 3: Solving Inequalities

Solve the inequality $\dfrac{4x+4}{2} > x$

We need to get rid of the fraction first by multiplying by 2

${4x+4} > 2x$

Then subtract $4x$

$4 > -2x$

Then divide by -2

$-2 < x$

Note: Remember the sign changes direction when multiplying or dividing by a negative number

### Example Questions

We solve this inequality by simply rearanging it to make k the subject,

\begin{aligned}7 - 3k &> -5k + 12 \\ 7 +2k&> 12 \\ 2k&>5 \\ k&>2.5\end{aligned}

Hence $k$ can take any value greater than $2.5$

We solve this inequality by simply rearanging it to make x the subject,

\begin{aligned}\dfrac{5x-1}{4} &> 3x \\ \\ 5x-1&> 12x \\ -1&>7x \\ x&<-\dfrac{1}{7}\end{aligned}

Hence $k$ can take any value less than $-\dfrac{1}{7}$

We solve this inequality by simply rearranging it to make $x$ the subject,

\begin{aligned}2x+5 &> 3x-2 \\ 7& > x \\ x&<7\end{aligned}

Hence $x$ can take any value less than7

We solve this inequality by simply rearranging it to make x the subject in the center of the inequality,

\begin{aligned}4-3x&\leq19 \\ -3x&\leq 15 \\ 3x&\geq -15\\ x&\geq-5\end{aligned}

Hence $x$ can take any value greater or equal to $-5$

We solve this inequality by simply rearranging it to make x the subject in the center of the inequality,

\begin{aligned}-5<2x&-3<10 \\ -2<2x&<13 \\ \\ -1

Hence $x$ can take any value grteater than $-1$ and less than $6.5$

Level 4-5

Level 4-5

Level 4-5

### Learning resources you may be interested in

We have a range of learning resources to compliment our website content perfectly. Check them out below.