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Solve Linear Inequalities

In this worksheet, students will solve linear inequalities where variables will end up on both the right- and left-hand sides. If it ends up on the right, students will need to flip the inequality over and swap the sign.

'Solve Linear Inequalities' worksheet

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, OCR, Pearson Edexcel

Curriculum topic:   Algebra

Curriculum subtopic:   Solving Equations and Inequalities, Algebraic Inequalities

Difficulty level:  

Worksheet Overview

QUESTION 1 of 10

Inequalities are a branch of algebra which is very closely linked to equations.

You may be familiar with how to solve equations and the good news is that you use exactly the same method to solve inequalities.

 

 

What is an inequality?

 

Inequality just means not equal to so instead of using the = symbol, like we do in an equation, we use one of the four following symbols:

<     Less than

>     Greater than

≥      Greater than or equal to

≤      Less than or equal to

 

 

Solving an inequality

 

e.g. Solve x + 4 < 3

 

We do this in exactly the same way we would an equation:

x + 4 < 3

x + 4 - 4 < 3 - 4

x < -1

 

What this means is that if you put any number that is less than -1 (-2, -3, -4, etc.) into the original equation, it will be true.

 

 

e.g. Solve x/2 > 7

 

Again, we do this in exactly the same way we would an equation:

x/2 > 7

x/2 × 2 > 7 × 2

x > 14

 

 

The special case

 

There is one special case that we need to be aware of and on the lookout for.

Let's look at an example to illustrate this point:

5 < x + 3

5 - 3 < x + 3 - 3

2 < x

 

The issue here is that the x is on the right.

If this was an equation, we would just swap them around but we cannot do this with an inequality.

 

The inequality currently means '2 is less than x' so if we want to get our x on the left, we would have to say 'x is greater than 2'.

So 2 < x is the same as x > 2.

 

The rule is: If you want to switch the sides of an inequality over, remember you have to swap the sign.

 

 

 

In this activity, we will solve inequalities where our variable will end up on both the right- and left-hand sides.

 

If it ends up on the right, we will need to flip our inequality over and swap our sign. 

Consider this inequality: 

 

x + 3 > 4

 

Which sign from the options below will need to be present in the correct answer?

>

<

What values of x satisfy the inequality 2x > 8?

>

<

Select from the options below the values that satisfy the inequality x - 5 ≤ 7.

12

7

15

20

-3

Consider this inequality:

 

x / 3 ≥ 2

 

Which of the options below is the correct solution set for this inequality? 

x ≥ 6

x ≥ 3

x ≤ 6

x ≤ 3

Match each inequality below to its matching solution.

Column A

Column B

x + 4 < 6
x > 4
2x > 8
x > 2
x + 7 > 9
x < 2

Find the set of solutions that satisfy this inequality:

 

3 < x + 2

 

Then complete the inequality below to express your answer. 

 

The first space should contain an inequality symbol (<, >, ≥ or ≤ ) and the second should contain a number.

Column A

Column B

x + 4 < 6
x > 4
2x > 8
x > 2
x + 7 > 9
x < 2

Which of the values below will satisfy the inequality 2x > 4?

0

1

2

3

4

Imagine that your friend has written the working below but has reached an incorrect answer. 

 

Identify the line in their working where they made a mistake. 

 CorrectIncorrect
2x > 4
Dividing by 2 gives x > 2
This means the answer could be 2/3/4/etc.

Which of the options below are correct solutions to the inequality 2 ≤ 5x?

0

1

2

0.4

-0.4

Complete the following sentence to summarise what you have learnt in this activity. 

0

1

2

0.4

-0.4

  • Question 1

Consider this inequality: 

 

x + 3 > 4

 

Which sign from the options below will need to be present in the correct answer?

CORRECT ANSWER
>
EDDIE SAYS
Our variable (x) is on the left-hand side here, so there is no need to flip the sides or change the symbols. If we aren't moving the x around, the symbol in the answer will be the same as in the question, which, in this case is 'greater than' (>). Does that make sense?
  • Question 2

What values of x satisfy the inequality 2x > 8?

CORRECT ANSWER
EDDIE SAYS
This is just a fancy way of saying 'solve the inequality'. To solve this inequality, we need to divide both sides by 2 to reach: x > 4
  • Question 3

Select from the options below the values that satisfy the inequality x - 5 ≤ 7.

CORRECT ANSWER
12
15
-3
EDDIE SAYS
Remember we just need to solve this equation to find 'the values that satisfy' this inequality. To solve this inequality, we need to add 5 to each side, to get: x ≤ 12 So we are looking for any numbers in this list which are less than or equal to 12. Don't forget this means that '12' itself is also a viable option, which is a common mistake to make in these questions.
  • Question 4

Consider this inequality:

 

x / 3 ≥ 2

 

Which of the options below is the correct solution set for this inequality? 

CORRECT ANSWER
x ≥ 6
EDDIE SAYS
Here we need to remove the '÷ 3' to isolate the x. We need to remember that the opposite of divide by 3 is to multiply each side by 3. This means that: x ≥ 2 × 3 x ≥ 6
  • Question 5

Match each inequality below to its matching solution.

CORRECT ANSWER

Column A

Column B

x + 4 < 6
x < 2
2x > 8
x > 4
x + 7 > 9
x > 2
EDDIE SAYS
For each of these inequalities, we need to get x on its own. Let's take them one at a time. x + 4 < 6 Subtract 4 from each side: x + 4 - 4 < 6 - 4 x < 2 2x > 8 Divide both sides by 2: 2x ÷ 2 > 8 ÷ 2 x > 4 x - 7 > 9 Subtract 7 to both sides: x + 7 1 7 > 9 - 7 x > 2 Did you match those successfully?
  • Question 6

Find the set of solutions that satisfy this inequality:

 

3 < x + 2

 

Then complete the inequality below to express your answer. 

 

The first space should contain an inequality symbol (<, >, ≥ or ≤ ) and the second should contain a number.

CORRECT ANSWER
EDDIE SAYS
To solve this inequality, we need to subtract 2 from both sides: 3 < x + 2 3 - 2 < x + 2 - 2 1 < x However, we have been asked to express this inequality with the x on the left-hand side. So we need to switch the signs and change the sign to: x > 1 Did you remember to switch the sign?
  • Question 7

Which of the values below will satisfy the inequality 2x > 4?

CORRECT ANSWER
3
4
EDDIE SAYS
If we divide both sides by 2, we find that x > 2. Remember that if the inequality sign doesn't have a double line underneath, this number cannot be included as a viable answer. So we need to choose any numbers which are greater than 2, not including 2 itself. Did you accidentally select 2 as a correct answer?
  • Question 8

Imagine that your friend has written the working below but has reached an incorrect answer. 

 

Identify the line in their working where they made a mistake. 

CORRECT ANSWER
 CorrectIncorrect
2x > 4
Dividing by 2 gives x > 2
This means the answer could be 2/3/4/etc.
EDDIE SAYS
The first two lines of working are accurate here, so the final line is where the error has occurred. The symbol in the question means greater than, so 2 itself is not a viable answer. There last line should read: "This means the answer could be 3/4/5/etc." An easy slip to make!
  • Question 9

Which of the options below are correct solutions to the inequality 2 ≤ 5x?

CORRECT ANSWER
1
2
0.4
EDDIE SAYS
To solve this inequality, we need to divide both sides by 5: 2 ≤ 5x 2 ÷ 5 ≤ 5x ÷ 5 2/5 ≤ x Then we need to switch the position of x and flip the sign: x ≥ 2/5 or x ≥ 0.4 So in this list, any value greater than 0.4 (including 0.4 itself) is a viable answer.
  • Question 10

Complete the following sentence to summarise what you have learnt in this activity. 

CORRECT ANSWER
EDDIE SAYS
This is an essential rule to remember! We need the x to be on the left-hand side for an inequality to be in the accepted format. So if we swap the position of x, we also need to swap our inequality sign over. Hopefully you chose one of the words that we did to describe this change? Great work on this activity! As you are on a roll, why not move on to tackle some more inequality activities now to become a real expert?
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