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Solving linear equations Inequalities on a number lineThis topic is relevant for:

Here we will learn about solving inequalities including how to solve linear inequalities, identify integers in the solution set and represent solutions on a number line.

There are also solving inequalities worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

Solving inequalities is where we calculate the values that an unknown variable can be in an inequality.

Solving inequalities is similar to solving equations, but where an equation has one unique solution, an inequality has a range of solutions.

To do this we need to balance the inequality in the same way as we would when solving an equation. Solutions can be integers or decimals, positive or negative numbers.

Solve

\[\begin{aligned}
2x+1&=9\\
2x&=8\\
x&=4
\end{aligned}\]

4 is the only solution to this equation.

2 × 4 + 1 = 9Solve

\[\begin{aligned}
2x+1&<9\\
2x&<8\\
x&<4
\end{aligned}\]

x can be any value that is less than 4

This **changes the direction** of the **inequality sign**.

E.g.

\[\begin{aligned}
1 – 2x&< 9\\
-2x&< 8\\
x&>-4
\end{aligned}\]

x can be any value that is greater than -4

In order to solve inequalities:

**Rearrange the inequality so that all the unknowns are on one side of the inequality sign.****Rearrange the inequality by dividing by the x coefficient so that ‘x’ is isolated.****Write your solution with the inequality symbol.**

Get your free solving inequalities worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEGet your free solving inequalities worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE**Solving inequalities** is part of our series of lessons to support revision on **inequalities**. You may find it helpful to start with the main inequalities lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

Solve

4x+6 < 26**Rearrange the inequality so that all the unknowns are on one side of the inequality sign.**

In this case you are subtracting ‘6’ from both sides.

\[\begin{aligned}
4x+6&<26\\
4x&<20\\
\end{aligned}\]

2**Rearrange the inequality by dividing by the x coefficient so that ‘x’ is isolated.**

In this case you need to divide both sides by 4 .

\[\begin{aligned}
4x+6&<26\\
4x&<20\\
x&<5
\end{aligned}\]

3**Write your solution with the inequality symbol.**

Any value less than 5 satisfies the inequality

Solve

5x-4\geq26**Rearrange the inequality so that all the unknowns are on one side of the inequality sign.**

In this case you need to add ‘4’ to both sides.

\[\begin{aligned}
5x-4&\geq26\\
5x&\geq30\\
\end{aligned}\]

**Rearrange the inequality by dividing by the x coefficient so that ‘x’ is isolated.**

In this case you need to divide both sides by 5 .

\[\begin{aligned}
5x-4&\geq26\\
5x&\geq30\\
x&\geq6\\
\end{aligned}\]

**Write your solution with the inequality symbol.**

x\geq6

Any value greater than or equal to 6 satisfies the inequality

Solve

3(x-4)\leq12**Rearrange the inequality so that all the unknowns are on one side of the inequality sign.**

Let’s start by expanding the bracket

3x-12\leq12Then we need to add ‘12’ to both sides.

\[\begin{aligned}
3x-12&\leq12\\
3x&\leq24\\
\end{aligned}\]

**Rearrange the inequality by dividing by the x coefficient so that ‘x’ is isolated.**

In this case you need to divide both sides by 3 .

\[\begin{aligned}
3x-12&\leq12\\
3x&\leq24\\
x&\leq8\\
\end{aligned}\]

**Write your solution with the inequality symbol.**

x\leq8

Any value less than or equal to 8 satisfies the inequality

Solve

5x - 6 > 2x + 15**Rearrange the inequality so that all the unknowns are on one side of the inequality sign.**

In this case you need to subtract ‘2x’ from both sides.

\[\begin{aligned}
5x-6&>2x+15\\
3x-6&>15\\
\end{aligned}\]

**Rearrange the inequality so that ‘x’s are on one side of the inequality sign and numbers on the other.**

In this case you need to add ‘6’ to both sides.

\[\begin{aligned}
3x-6&>15\\
3x&>21\\
\end{aligned}\]

**Rearrange the inequality by dividing by the x coefficient so that ‘x’ is isolated.**

In this case you need to divide both sides by 3 .

\[\begin{aligned}
3x&>21\\
x&>7\\
\end{aligned}\]

**Write your solution with the inequality symbol.**

x > 7

Any value greater than 7 satisfies the inequality

Solve

\frac{x+3}{5}<2**Rearrange the inequality to eliminate the denominator.**

In this case you need to multiply both sides by 5 .

\[\begin{aligned}
\frac{x+3}{5}<2\\
x+3&<10\\
\end{aligned}\]

**Rearrange the inequality so that ‘x’s are on one side of the inequality sign and numbers on the other.**

In this case you need to subtract ‘3’ from both sides

\[\begin{aligned}
\frac{x+3}{5}<2\\
x+3&<10\\
x&<7\\
\end{aligned}\]

**Write your solution with the inequality symbol.**

x < 7

Any value less than 7 satisfies the inequality

Solve

6x+1\geq4**Rearrange the inequality so that all the unknowns are on one side of the inequality sign.**

In this case you need to subtract ‘1’ from both sides.

\[\begin{aligned}
6x+1&\geq4\\
6x&\geq3\\
\end{aligned}\]

**Rearrange the inequality by dividing by the x coefficient so that ‘x’ is isolated.**

In this case you need to divide both sides by 6 .

\[\begin{aligned}
6x+1&\geq4\\
6x&\geq3\\
x&\geq\frac{3}{6}\\
\end{aligned}\]

This can be simplified to \frac{1}{2} or the decimal equivalent.

**Write your solution with the inequality symbol.**

x\geq\frac{1}{2}

Any value less than \frac{1}{2} satisfies the inequality

Represent the solution on a number line

2x - 7 < 5** ‘x’**s are on one side of the inequality sign and numbers on the other.

In this case you need to add ‘7’ to both sides.

\[\begin{aligned}
2x-7&<5\\
2x&<12\\
\end{aligned}\]

**Rearrange the inequality by dividing by the x coefficient so that ‘x’ is isolated.**

In this case you need to divide both sides by 2 .

\[\begin{aligned}
2x-7&<5\\
2x&<12\\
x&<6\\
\end{aligned}\]

**Represent your solution on a number line.**

x <6

Any value less than 6 satisfies the inequality. An open circle is required at 6 and the value lower than 6 indicated with an arrow.

Solve

1 - 2x <7** ‘x’**s are on one side of the inequality sign and numbers on the other.

In this case you need to subtract ‘1’ from both sides.

\[\begin{aligned}
1 – 2x &<7 \\
– 2x&<6 \\
\end{aligned}\]

**Rearrange the inequality by dividing by the x coefficient so that ‘x’ is isolated.**

In this case you need to divide both sides by negative 2 .

\[\begin{aligned}
1 – 2x&<7 \\
– 2x&<6\\
x&◯ -3 \\
\end{aligned}\]

**Change the direction of the inequality sign.**

Because you divided by a negative number, you also need to change the direction of the inequality sign.

x > -3List the integer values that satisfy

3<x+1\leq8**Rearrange the inequality so that all the unknowns are on one side of the inequality sign.**

In this case you need to subtract ‘1’ from each part.

\[\begin{aligned}
3<x+1\leq8\\
2<x\leq7\\
\end{aligned}\]

**List the integer values satisfied by the inequality.**

2<x\leq7

2 is not included in the solution set. 7 is included in the solution set. The integers that satisfy this inequality are:

3, 4, 5, 6, 7List the integer values that satisfy

7\leq4x\leq20**Rearrange the inequality so that all the unknowns are on one side of the inequality sign.**

In this case you need to divide each part by ‘4’ .

\[\begin{aligned}
7\leq4 x \leq20\\
\frac{7}{4}\leq x \leq5\\
\end{aligned}\]

**List the integer values satisfied by the inequality.**

\frac{7}{4} \leq x \leq5

\frac{7}{4} is included in the solution set but it is not an integer. The first integer higher is ‘2’ . 5 is also included in the solution set. The integers that satisfy this inequality are:

2, 3, 4, 5List the integer values that satisfy

-3<2x+5\leq7**Rearrange the inequality so that all the unknowns are on one side of the inequality sign.**

In this case you need to subtract ‘5’ from each part.

\[\begin{aligned}
-3<2x+5\leq7\\
-8<2x\leq2\\
\end{aligned}\]

**Rearrange the inequality so that ‘x’ is isolated. In this case you need to divide each part by 2.**

\[\begin{aligned}
-3<2x+5\leq7\\
-8<2x\leq2\\
-4<x\leq1\\
\end{aligned}\]

**Represent the solution set on a the number line**

-4<x\leq1

-4 is not included in the solution set so requires an open circle. 1 is included in the solution set so requires a closed circle. Put a solid line between the circles to indicate all the values that satisfy the solution set.

**Solutions as inequalities**

Not including the inequality symbol in the solution is a common mistake. An inequality has a range of values that satisfy it rather than a unique solution so the inequality symbol is essential

E.g.

When solving x + 3 < 7 giving a solution of ‘4’ or ‘x = 4’ is incorrect, the answer must be written as an inequality ‘x < 4’

**Balancing inequalities**

Errors can be made with solving equations and inequalities by not applying inverse operations or not balancing the inequalities. Working should be shown step-by-step with the inverse operations applied to both sides of the inequality.

E.g.

When solving x + 3 < 7 , adding ‘3’ to both sides rather than subtracting ‘3’ from both sides

1. Solve 3x+7 < 31

3x < 24

x > 8

8

x < 8

\begin{aligned}
3x+7&<31\\
3x&<24\\
x&<8
\end{aligned}

2. Solve 4x-3\geq25

x < 7

x\geq7

x > 7

7

\begin{aligned}
4x-3&\geq25\\
4x&\geq28\\
x&\geq7\\
\end{aligned}

3. Solve 2(x-5)\leq8

x < 9

x\leq-1

x\leq9

9

\begin{aligned}
2(x-5)&\leq8\\
2x-10&\leq8\\
2x&\leq18\\
x&\leq9\\
\end{aligned}

4. Solve 6x – 5 > 4x + 1

x > 3

x < 3.5

x\leq3

3

\begin{aligned}
6x-5&>4x+1\\
2x-5&>1\\
2x&>6\\
x&>3\\
\end{aligned}

5. Solve \frac{x-4}{2}>6

x < 4

x >16

x\leq3

16

\begin{aligned}
\frac{x-4}{2}>6\\
x-4&>12\\
x&>16\\
\end{aligned}

6. Solve 8x+1\geq3

x\geq4

x\leq4

\frac{1}{4}

x\geq\frac{1}{4}

\begin{aligned}
8x+1&\geq3\\
8x&\geq2\\
x&\geq\frac{2}{8}\\
x&\geq\frac{1}{4}\\
\end{aligned}

7. Represent the solution on a number line 5x – 2 < 28

\begin{aligned}
5x-2&<28\\
5x&<30\\
x&<6\\
\end{aligned}

An open circle is required and all values less than 6 indicated.

8. Solve 2 – 3x > 14

x > 4

x < 4

-x > 4

x<\frac{16}{3}

\begin{aligned}
2- 3x&>14\\
-3x&>12 \\
x&< 4\\
\end{aligned}

Change the direction of the inequality sign as you have divided by a negative number

9. List the integer values that satisfy 2<x+3\leq5

0, 1, 2

2, 3, 4, 5

3, 4, 5

-1, 0, 1, 2

\begin{aligned}
2<x+3\leq5\\
-1<x\leq2\\
\end{aligned}

-1 is not included in the solution set as is greater than -1 .

2 is included in the solution set as x is less than or equal to 2 .

10. List the integer values that satisfy 4\leq3x\leq21

1, 2, 3, 4, 5, 6, 7

2, 3, 4, 5, 6

1, 2, 3, 4, 5, 6

2, 3, 4, 5, 6, 7

\begin{aligned}
4\leq3 x \leq21\\
\frac{4}{3}\leq x \leq7\\
\end{aligned}

The first integer greater than \frac{4}{3} is 2 .

7 is included in the solution set as x is less than or equal to 7 .

11. List the integer values that satisfy -4<3x+2\leq5

-2, -1, 0, 1

-1, 0

-1, 0, 1

-3, -2, -1, 0, 1, 2, 3, 4, 5

\begin{aligned}
-4<3x+2\leq5\\
-6<3x\leq3\\
-2<x\leq1\\
\end{aligned}

-2 is not included in the solution set as x is greater than -2 .

1 is included in the solution set as x is less than or equal to 1 .

1. John’s solution to 2x + 5 > 17 is shown on the number line

Is John’s solution correct?

Explain your reasoning.

**(2 marks)**

Show answer

Correctly solves the inequality

**(1)**

No, correct solution is x > 6

Indicates ‘no’ with correct reason or represents correct inequality on the number line

**(1)**

2. (a) Solve 4x+1\leq3x-2

(b) Represent your solution to (a) on the number line

**(4 marks)**

Show answer

(a)

Correct attempt at solving, for example eliminating ‘x’ . x+1\leq-2

**(1)**

Correct solution x\leq-3

**(1)**

(b)

‘-3’ or their value indicated on the number line with a closed circle

**(1)**

Correct inequality or their inequality shown on the number line with aa closed circle and values on the left side of the circle indicated with an arrow.

**(1)**

3. (a) Solve 5x – 1 > 9

(b) Write down the smallest integer that satisfies 5x – 1 > 9

**(3 marks)**

Show answer

(a)

Correct attempt at solving: 5x > 10

**(1)**

Correct solution: x > 2

**(1)**

(b)

Correct solution: 3

**(1)**

You have now learned how to:

- Solve inequalities
- Represent your solutions on a number line
- List integer values that satisfy your inequality

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