Solving inequalities

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Solving linear equations Inequalities on a number line

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GCSE Maths Algebra Inequalities

Solving Inequalities

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.

What is solving inequalities?

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.

What is solving inequalities?

Solving an equation

Solve

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

$4$ is the only solution to this equation.

2 \times 4 + 1 = 9

Solving an inequality

Solve

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

$x$ can be any value that is less than $4$

Multiplying and dividing by a negative number

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 $-4$

How to solve inequalities

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.

Explain how to solve inequalities

Solving inequalities worksheet

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

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Solving inequalities worksheet

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

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Related lessons on inequalities

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:

Solving linear inequalities examples

Example 1: solving linear inequalities

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.

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

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

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

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

3Write your solution with the inequality symbol.

x < 5

Any value less than $5$ satisfies the inequality

Example 2: solving linear inequalities

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

Example 3: solving linear inequalities with brackets

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\leq12

Then 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

Example 4: solving linear inequalities with unknowns on both sides

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

Example 5: solving linear inequalities with fractions

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

Example 6: solving linear inequalities – non integer solutions

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

Example 7: solving linear inequalities and representing solutions on a number line

Represent the solution on a number line

2x - 7 < 5

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 $‘7’$ 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.

Example 8: solving linear inequalities with negative x coefficients

Solve

1 - 2x <7

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 $‘1’$ 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 > -3

Example 9: solving linear inequalities and listing integer values that satisfy the inequality

List 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, 7

Example 10: solving linear inequalities and listing integer values that satisfy the inequality

List 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, 5

Example 11: solving linear inequalities and representing the solution on a number line

List 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.

Common misconceptions

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

Practice solving inequalities questions

3x < 24

x > 8

8

x < 8

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

x < 7

x\geq7

x > 7

7

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

x < 9

x\leq-1

x\leq9

9

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

x > 3

x < 3.5

x\leq3

3

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

x < 4

x >16

x\leq3

16

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

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}

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

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

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

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 $2$ .

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 $7$ .

-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 $1$ .

Solving inequalities GCSE questions

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)

Learning checklist

You have now learned how to:

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

The next lessons are

Still stuck?

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