 # Solve Two-Stage Inequalities

In this worksheet, students will find solution sets for inequalities with two elements by applying inverse operations to isolate the target variable. Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, OCR, Pearson Edexcel

Curriculum topic:   Algebra

Curriculum subtopic:   Solving Equations and Inequalities, Algebraic Equations

Difficulty level:   ### QUESTION 1 of 10

Finding the solution set for inequalities can be quite straight forward, as long as we treat them exactly the same as equation and follow these rules:

1) Eliminate terms by applying the inverse (opposite) operation.

2) Always do the same things to both sides.

The main difference between an equation and an inequality (apart from the signs) is that we cannot just move the x to the other side if it is not where we want it to be.

In equations, 1 = x and x = 1 are the same thing.

In inequalities, if we want to change the side x is on, we must change the inequality sign as well.

So 1 < x would be the same as x > 1.

e.g. Find the solution set for 3x + 5 ≤ 2.

We are trying to get x by itself (or isolate it).

So we first need to get rid of the + 5, we do this by using the inverse - 5.

3x + 5 - 5 ≤ 2 - 5

3x ≤ -3

Next, we have to eliminate the × 3, we do this by ÷ 3:

3x ÷ 3 ≤ -3 ÷ 3

x ≤ -1

e.g. Find the solution set for -2 > 3x + 4.

We are trying to get x by itself, so first we need to get rid of the + 5, we do this by using - 5:

-2 - 4 > 3x + 4 - 4

-6 > 3x

Next, we have to eliminate the × 3, we do this by using ÷ 3:

- 6 ÷ 3 > 3x ÷ 3

-2 > x

Lastly, this inequality has its x on the right and it needs to be on the left.

To manage this, we also have to switch the inequality:

x < -2

In this activity, we will find solution sets for inequalities with two elements by isolating our target term on the left-hand side or flipping the side and switching the inequality sign if our variable is on the left.

Some of the working for these questions can be lengthy, so you may want to have a pen and paper handy so you can compare your own working to that of our maths teacher.

Which inequality sign will be present in the solution set for 2x + 1 > 5?

>

<

Which values of x will satisfy the inequality 3x - 7 < 2?

The first space should contain an inequality sign (i.e. <, >, &le;, &ge;) and the second should contain a number

>

<

Which of the following values satisfy the inequality 2x - 7 ≤ 13?

1

5

9

10

15

20

Which of these solution sets is correct for the inequality 2x - 3 ≤ 10?

x ≤ 6.5

x ≤ 7

x ≥ 6.6

Match each inequality below to its solution set.

## Column B

2x + 3 > 5
x > 3
3x - 2 > 7
x > 1.25
5 < 3x - 1
x > 2
4x - 5 > 0
x > 1

Which values of x will satisfy the inequality 5 < 3x - 1?

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

## Column B

2x + 3 > 5
x > 3
3x - 2 > 7
x > 1.25
5 < 3x - 1
x > 2
4x - 5 > 0
x > 1

Which of the following values satisfy the inequality 2x - 5 > 11?

1

5

8

10

15

20

Imagine that your friend has been working out the solution set to this inequality:

3x - 4 > 2

They have found their answer but it is not correct.

They have recorded their working in the three lines below.

In which line have they made an error?

3x - 4 > 2

3x > -2

3x > -2/3

What is the smallest whole number solution for the inequality 3 < 4x + 2?

What is the largest square number that satisfies the inequality 3x - 7 < 23?

• Question 1

Which inequality sign will be present in the solution set for 2x + 1 > 5?

>
EDDIE SAYS
The sign in an inequality will stay the same when we are finding our solution sets, unless the variable (x) is on the right when we have finished. This is not the case in this example, as our x is already on the left and will stay there as we calculate the solution set. Does that make sense?
• Question 2

Which values of x will satisfy the inequality 3x - 7 < 2?

The first space should contain an inequality sign (i.e. <, >, &le;, &ge;) and the second should contain a number

EDDIE SAYS
This is just a fancy way of saying 'find the solution set'. Unless the question asks for something else (such as 'what integers satisfy...'), we just need to work this out in the same way we would with an equation. Let's start by applying the inverse of - 7 which is + 7: 3x - 7 + 7 < 2 + 7 3x < 9 Now we need to apply the inverse of × 3 which is ÷ 3: 3x ÷ 3 < 9 ÷ 3 x < 3 Did you type the inequality sign and number into the correct blanks?
• Question 3

Which of the following values satisfy the inequality 2x - 7 ≤ 13?

1
5
9
10
EDDIE SAYS
Our first step here is to find the solution set by applying the inverse operations to isolate x: 2x - 7 ≤ 13 2x - 7 + 7 ≤ 13 + 7 2x ≤ 20 2x ÷ 2 ≤ 20 ÷ 2 x ≤ 10 This means that any number which is less than or equal to 10 will satisfy this inequality. Remember that 10 itself is a valid answer, as this inequality uses the ≤ sign rather than <.
• Question 4

Which of these solution sets is correct for the inequality 2x - 3 ≤ 10?

x ≤ 6.5
EDDIE SAYS
A common mistake here is to think that the solution set must use a whole number. If we solve this inequality as an equation by applying the inverse operations to isolate x, our working would be: 2x - 3 ≤ 10 2x - 3 + 3 ≤ 10 + 3 2x ≤ 13 2x ÷ 2 ≤ 13 ÷ 2 x ≤ 6.5 There is nothing wrong at all with our answer using a decimal. Remember that our inequality sign will stay the same unless the x is on the right.
• Question 5

Match each inequality below to its solution set.

## Column B

2x + 3 > 5
x > 1
3x - 2 > 7
x > 3
5 < 3x - 1
x > 2
4x - 5 > 0
x > 1.25
EDDIE SAYS
Did you remember the rules? 1) Solve it the same way we would an equation. 2) Keep the sign the same, unless the x is on the right. If we apply these accurately, we find that only one option requires us to flip the sign as the x finishes on the right. Let's take a closer look at this one. 5 < 3x - 1 5 + 1 < 3x - 1 + 1 6 < 3x 6 ÷ 3 < 3x ÷ 3 2 < x We want our x to finish on the left, so we need to switch the side of x and switch our inequality sign. Think about it like this, our original inequality (2 < x) says that "2 is less than x" so when we flip it to x > 2 this means that "x is greater than 2". These two statements actually mean the same thing!
• Question 6

Which values of x will satisfy the inequality 5 < 3x - 1?

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

EDDIE SAYS
Did you notice that our x ended up on the right here? Remember that this means we must switch the inequality sign. Here is our working for this one: 5 < 3x - 1 5 + 1 < 3x - 1 + 1 6 < 3x 6 ÷ 3 < 3x ÷ 3 2 < x --> x > 2
• Question 7

Which of the following values satisfy the inequality 2x - 5 > 11?

10
15
20
EDDIE SAYS
Our first step here is to find the solution set by treating this like an equation and isolating x by applying inverse operations: 2x - 5 > 11 2x - 5 + 5 > 11 + 5 2x > 16 2x ÷ 2 > 16 ÷ 2 x > 8 This means that any number which is more than but not equal to 8 is a viable answer. Hopefully you didn't select 8 itself? For this to be a possible answer, we would need to have seen the ≥ sign rather than the > sign present.
• Question 8

Imagine that your friend has been working out the solution set to this inequality:

3x - 4 > 2

They have found their answer but it is not correct.

They have recorded their working in the three lines below.

In which line have they made an error?

3x > -2
EDDIE SAYS
Did you spot their error? They have broken one of the golden rules for solving inequalities - in line 2 they have not applied the . The opposite of - 4 is + 4. If we add 4 to the 2 on the right, it would give 6 not -4. Your friend has subtracted 4 rather than added it - an easy mistake to make!
• Question 9

What is the smallest whole number solution for the inequality 3 < 4x + 2?

1
EDDIE SAYS
Our first step here is to find the solution set by treating this as an equation and applying the inverse operations to isolate x: 3 < 4x + 2 3 - 2 < 4x + 2 - 2 1 < 4x 1 ÷ 4 < 4x ÷ 4 1/4 < x --> x > 1/4 or x > 0.25 The other part of this question asks for the smallest whole number solution. This is a more complicated way of asking, "What is the smallest number that meets this criteria?" We have found that x must be greater than 0.25. What is the first whole number which is greater than 0.25? That's right, it's 1.
• Question 10

What is the largest square number that satisfies the inequality 3x - 7 < 23?

9
EDDIE SAYS
Again, our first step here is to find the solution set. 3x - 7 < 23 3x - 7 + 7 < 23 + 7 3x < 30 3x ÷ 3 < 30 ÷ 3 x < 10 This time we are looking for the last square number that is smaller than 10. Remember that square numbers are numbers which are the product of multiplying a number by itself. The closest we can get to 10 is 9 which is 3 × 3. Congratulations for completing this activity! Why not level up and try another activity in this series if you are feeling confident after this one?
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