# Solving Inequalities

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## Solving Inequalities

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.

Make sure you are happy with the following topics before continuing.

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## Type 1: Listing values

$x$ is an integer such that $-1\leq x \lt 4$. List all numbers that satisfy this inequality.

For such questions you need consider if the inequalities are inclusive or strict, in this case we have,

$x$ takes any value greater then or equal to $-1$       and        $x$ takes any value less than $4$

Hence, the integers that satisfy the inequality are: $-1,0,1,2,3$

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## Type 2:Solving Inequalities Basic

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

Firstly, add $4$ to both sides of the inequality to get,

\begin{aligned}(\textcolor{maroon}{+4})\,\,\,\,\,\,\,\,\, 5a -4 &\gt 2a+8 \\ 5a &\gt 2a+12 \end{aligned}

Then, subtract $2a$ from both sides to get,

\begin{aligned}(\textcolor{maroon}{-2a})\,\,\,\,\,\,\,\,\, 5a &\gt 2a+12 \\ 3a &\gt 12 \end{aligned}

Finally, divide both sides by $3$ to get,

\begin{aligned}(\textcolor{maroon}{\div 3})\,\,\,\,\,\,\,\,\, 3a &\gt 12 \\ a &\gt 4 \end{aligned}

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## Type 3: Solving Inequalities 2 signs

Solve the inequality $5 \lt 2x-3 \lt 13$

Firstly, add $3$ to each side of the inequality,

(Remember what you do to one side you do to all sides,  even if there are $3$ sides), to get

\begin{aligned}(\textcolor{maroon}{+3})\,\,\,\,\,\,\,\,\, 5& \lt 2x-3 \lt 13 \\ 8 &\lt 2x \lt 16 \end{aligned}

Finally, divide both sides by $2$ to get,

\begin{aligned}(\textcolor{maroon}{\div 2})\,\,\,\,\,\,\,\,\, 8 \lt 2x& \lt 16 \\ 4 \lt x& \lt 8 \end{aligned}

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## Type 4: Multiplying and Dividing by a Negative Number

When rearranging an inequality, you are performing the same operation to both sides of the inequality without changing it (just like as you would with an equation) but with one exception:

If you multiply or divide by a negative number, then the inequality sign changes direction

For example, if we have to solve the inequality $-2x \gt 4$, we have to divide both sides by $-2$,

\begin{aligned}(\textcolor{maroon}{\div -2})\,\,\,\,\,\,\,\,\, -2x &\gt 4 \\ x &\lt -2 \end{aligned}

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## Example

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

[3 marks]

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

${4x+4} > 2x$

Then subtract $4x$

$4 > -2x$

Then divide by $-2$

$-2 < x$

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

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## Solving Inequalities Example Questions

Question 1: Solve the inequality $7 - 3k > -5k + 12$

[2 marks]

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

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

Hence $k$ can take any value greater than $\dfrac{5}{2}$

Gold Standard Education

Question 2: Solve the inequality $\dfrac{5x-1}{4} > 3x$

[3 marks]

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We solve this inequality by simply rearranging 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 $-\dfrac{1}{7}$

Gold Standard Education

Question 3: Solve the inequality $2x+5 > 3x-2$

[2 marks]

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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 than $7$

Gold Standard Education

Question 4: Find the range of values that satisfy the inequality, $4-3x\leq19$

[3 marks]

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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 $-5$

Gold Standard Education

Question 5: Find the range of values that satisfy the inequality $-5<2x-3<10$

[2 marks]

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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 greater than $-1$ and less than $\dfrac{13}{2}$

Gold Standard Education

## Solving Inequalities Worksheet and Example Questions

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