 # Solve Inequalities (on a number line)

In this worksheet, students will learn and apply the rules for illustrating solution sets for inequalities, as well as identifying the correct illustrations to represent inequalities. 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:   ### QUESTION 1 of 10

When we are working with inequalities, a common question to be asked is to state all the numbers that satisfy an inequality.

We can either do this by giving a solution set or by drawing the numbers on a number line.

In this activity, we are going to focus on the second option here of representing inequalities on a number line.

e.g. Illustrate the inequality 3x – 4 > 2 on a number line.

The first step here is to solve the inequality:

3x – 4 + 4 > 2 + 4

3x > 6

3x ÷ 3 > 6 ÷ 3

x > 2

Now we have our solution set, we can draw this on a number line.

As the inequality uses a greater than sign, we use an unshaded circle to show the value of 2.

We know that the viable solutions must be greater than 2, so we draw an arrow to the right, like this: Let's try another now.

e.g. Illustrate the inequality 2x + 1 ≤ 5 on a number line.

The first step here is to solve the inequality:

2x + 1 ≤ 5

2x + 1 -1 ≤ 5 - 1

2x ≤ 4

2x ÷ 2 ≤ 4 ÷ 2

x ≤ 2

Now we have our solution set, we can draw this on a number line.

As the inequality uses a less than or equal to sign, we use a shaded circle to show the value of 2.

We know that the viable solutions must be less than or equal 2, so we draw an arrow to the left, like this: Let's practise one more before we try the activity; this time using a inequality with three parts.

e.g. Illustrate the inequality -1 ≤ 2x – 3 < 5 on a number line.

The first step here is to solve the inequality:

-1 + 3 ≤ 2x – 3 + 3 < 5 + 3

2 ≤ 2x < 8

2 ÷ 2  ≤ 2x ÷ 2  < 8 ÷ 2

1 ≤ x < 4

Now we have our solution set, we can draw this on a number line.

We can see that the viable solutions must be between 1 and 4.

The circle at 1 has to be shaded (as it has a ≤ sign) and the one at 4 (as it has a < sign) has to be unshaded, like this: In this activity, you will revise and apply the rules for illustrating solution sets for inequalities as well as identifying the correct illustrations to represent inequalities.

You will not be required to actually draw any number lines in this activity to reach the correct answers, unless you choose to.

However, once you have completed this activity, you may want to go back through each question and draw number lines to represent some of the inequalities shown here so that you can practise this skill further.

Read the statement below then choose the correct word to ensure it is true.

When illustrating an inequality, we use an ______ circle for > and < signs.

Read the statement below then choose the correct word to ensure it is true.

When illustrating an inequality, we use an ______ circle for ≥ and ≤.

What is the solution set for 2x + 1 < 5?

In the first blank you should type an inequality symbol (<, >, ≤ or ≥), and in the second you should type a number.

Which of the number lines below show the correct solution for 2x + 1 < 5?  What is the solution set for 3x – 5 ≥ -8?  Which of the number lines below show the correct solution for 3x – 5 ≥ -8? A

B

C

D

What is the solution set for 5x + 3 > 2x + 9?

In the first blank you should type an inequality symbol (<, >, ≤ or ≥), and in the second you should type a number.

A

B

C

D

Which of the number lines below show the correct solution for 5x + 3 > 2x + 9? A

B

C

D

Match each solution set beloe to its correct illustration on the number line.

## Column B

x > 4

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x < -1

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x ≥ -1

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Match each inequality below with its correct solution set.

## Column B

2x – 5 < 3
x > -2
2x – 5 > -7
x > 4
2x – 5 > -9
x < -1
• Question 1

Read the statement below then choose the correct word to ensure it is true.

When illustrating an inequality, we use an ______ circle for > and < signs.

EDDIE SAYS
Did you recall this information from the Introduction? When we are dealing with < or > signs, we cannot use the number itself as part of our solution set. We show this by not shading the circle on our number line.
• Question 2

Read the statement below then choose the correct word to ensure it is true.

When illustrating an inequality, we use an ______ circle for ≥ and ≤.

EDDIE SAYS
This is the opposite situation to our previous question. When we are dealing with ≥ or ≤ signs, we must use the number itself as part of our solution set. We show this by shading the circle on our number line. Remember these two key facts to support you in the rest of this activity.
• Question 3

What is the solution set for 2x + 1 < 5?

In the first blank you should type an inequality symbol (<, >, ≤ or ≥), and in the second you should type a number.

EDDIE SAYS
We need to solve this in the same way we would solve an equation, by applying inverse operations and isolating x. We subtract 1 from both sides: 2x + 1 < 5 2x + 1 - 1 < 5 - 1 Then we divide both sides by 2: 2x < 4 2x ÷ 2 < 4 ÷ 2 This will give us the solution: x < 2 Did you type this symbol and number into the correct spaces?
• Question 4

Which of the number lines below show the correct solution for 2x + 1 < 5? EDDIE SAYS
We solved this inequality in the previous question to find the solution set of x <2. Because the inequality is <, we need to use an unshaded circle on our number line, with an arrow pointing to the left to show less than. Which image accurately shows this? It's the first one, as the second shows a solution of x > 2 instead.
• Question 5

What is the solution set for 3x – 5 ≥ -8?

EDDIE SAYS
Again, we solve this the same way as we would an equation, by isolating x and applying inverse operations. Add 5 to both side: 3x – 5 ≥ -8 3x – 5 + 5 ≥ -8 + 5 Divide both sides by 3: 3x ≥ -3 3x ÷ 3 ≥ -3 ÷ 3 So our solution set is: x ≥ -1 Remember that our inequality always remains the same, unless our x is on the right-hand side.
• Question 6

Which of the number lines below show the correct solution for 3x – 5 ≥ -8? C
EDDIE SAYS
In the previous question, we found that x ≥ -1 is the solution set for this inequality. From this, we know we are looking for a circle at -1 which is shaded since a ≥ sign is present. The arrow should be pointed to the right, to show that viable solutions need to be greater than this number. Did you locate the correct number line?
• Question 7

What is the solution set for 5x + 3 > 2x + 9?

In the first blank you should type an inequality symbol (<, >, ≤ or ≥), and in the second you should type a number.

EDDIE SAYS
The only difference here is that we have unknowns on both sides. This means that our first step need to be to eliminate one of them. Here is our working: 5x + 3 > 2x + 9 5x + 3 - 2x > 2x - 2x + 9 3x + 3 > 9 3x + 3 - 3 > 9 - 3 3x > 6 3x ÷ 3 > 6 ÷ 3 x > 2 Did you type the correct inequality symbol and number into those spaces?
• Question 8

Which of the number lines below show the correct solution for 5x + 3 > 2x + 9? A
EDDIE SAYS
In the previous question, we found that the solution set for this inequality is x > 2. So we are looking for a circle at 2 which is unshaded, as we have a > sign present. Our arrow needs to point to the right to indicate that only solutions greater than are viable solutions.
• Question 9

Match each solution set beloe to its correct illustration on the number line.

## Column B

x > 4

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x < -1

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x ≥ -1

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EDDIE SAYS
With these, you can either start by looking at the inequality itself as there’s only one that uses a greater than or equal to symbol. This means that we can match this with the only shaded circle. Alternatively, we can match the number first. The first solution set needs to show a circle at 4, so this fact should make it easier to spot. How did you get on here?
• Question 10

Match each inequality below with its correct solution set.

## Column B

2x – 5 < 3
x > 4
2x – 5 > -7
x < -1
2x – 5 > -9
x > -2
EDDIE SAYS
While this looks quite difficult at first glance, we can just follow the same rules of solving these as equations by applying inverse operations. Let's look at the first one together: 2x – 5 < 3 2x – 5 + 5 < 3 + 5 2x < 8 2x ÷ 2 < 8 ÷ 2 x < 4 Can you solve and match the other two pairs independently? Great job completing this activity! If you want to practise this skill further, why not go back through this activity and draw number lines to illustrate any inequalities which do not currently have a visual. Why not try our Level 2 activity on solving inequalities if you are ready for the challenge?
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