 # Solve Complex Inequalities

In this worksheet, students will practise solving multi-stage inequalities, where they may need to switch the sign at the end or may involve more than two elements. 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

Solving inequalities can be simple if we follow a couple of golden rules:

1) Treat them in the same way as an equation.

2) If you want to switch the x in a solution set from the right to the left side, you have to switch the sign (i.e. 1 > x is the same as x > 1).

A quick word on the language, when we are dealing with inequalities, we aren't finding a solution (this word applies to equations only), we are finding a solution set.

e.g. Find the solutions for 2x + 1 < 5x + 3.

Step 1:

We have a value of x on both sides, so we have to eliminate one of them.

Always get rid of the smaller one, as this helps to avoid negatives.

3x + 1 < 5x + 3

3x + 1 - 3x < 5x + 3 - 3x

1 < 2x + 3

Step 2:

Now let's isolate our x:

1 < 2x + 3

1 - 3 < 2x + 3 - 3

-2 <  2x

-2 ÷ 2<  2x ÷ 2

-1 < x

Step 3:

Switch the inequality over if necessary:

x > -1

Now let's look at another example where our inequality has three parts.

e.g. Find the solutions for -5 ≤ 3x + 1 ≤ 7.

The key thing to remember is that we have to do the same to all parts of the inequality:

-5 ≤ 3x + 1 ≤ 7

-5 - 1 ≤ 3x + 1 - 1 ≤ 7 - 1

-6 ≤ 3x ≤ 6

-6 ÷ 3≤ 3x ÷ 3 ≤ 6 ÷ 3

-2 ≤ x ≤ 2

In this activity, we will find solutions for complex inequalities where we may need to switch the sign at the end or may involve more than two elements.

Our working for these questions can be long and complex, so make sure you have a pen and paper handy to record yours and then compare it to what our maths teacher has written as an example solution.

Consider the inequality below:

2x + 1 < 5x + 3

What is the first step we should take to solve this inequality?

Subtract the terms which contain x

Ignore the terms which contain x

Combine the terms which contain x

When we are solving a three-part inequality, we must...

Do the same to both sides

Do the same to all three parts

Treat each part as a separate element

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

6x - 4 > 3x + 11

x ≤ 5

x < 5

x > 5

x ≥ 5

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

19 ≤ 4x + 3 < 23

4

4.2

4.6

5

Find the solution set for this inequality:

5x + 1 < 3x + 5

Find the solution set for this inequality:

3 < 3x < 9

Don't put any spaces in your answer or you may be marked incorrectly.

Match each inequality below with its solution set.

## Column B

6x + 1 > 2x + 9
x > 2
2x + 2 < 14 - x
x < 4
x + 8 > 7x - 4
x < 2

Match each inequality below with its solution set.

## Column B

7 < 3x + 1 < 16
-1 < x < 3
-7 < 4x - 3 < 9
2 < x < 5
-9 < 5x + 6 < 1
-3 < x < -1

Find the solution set for this inequality:

4x - 7 > 6x - 13

Find the solution set for this inequality:

7 < 4x - 5 < 17

Don't put any spaces in your answer and express any number less than 1 as decimals or you may be marked incorrectly.

• Question 1

Consider the inequality below:

2x + 1 < 5x + 3

What is the first step we should take to solve this inequality?

Combine the terms which contain x
EDDIE SAYS
The first challenge to address with this inequality is that we have values of x on both sides of the inequality sign. In order to get this into a form we can deal with, we need to combine (or eliminate) the x terms. Review the Introduction now to get those 3 steps locked down in your mind before you proceed.
• Question 2

When we are solving a three-part inequality, we must...

Do the same to all three parts
EDDIE SAYS
It's a common mistake with these questions, to think we need to do something to both sides. This is what we have to do with equation, and doesn't work for an inequality. In a three-part inequality or more, we have to do the same to all the parts of the inequality.
• Question 3

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

6x - 4 > 3x + 11

x > 5
EDDIE SAYS
1) Let's start by eliminating one of our x terms, 3x is the smallest so let's eliminate that. We can subtract 3x from both sides: 3x - 4 > 11 2) Now we need to get x on its own or isolate it. Add 4 to both sides: 3x > 15 x ÷ 3 > 15 ÷ 3 x > 5 3) As the larger value of x is on the right already here, we aren't going to have to flip our signs at the end. This means the symbol in the middle will stay the same. Did you follow those steps correctly?
• Question 4

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

19 ≤ 4x + 3 < 23

4
4.2
4.6
EDDIE SAYS
Firstly, we need to solve this inequality. Remember that we must do the same thing to all three parts of our inequality. 19 ≤ 4x + 3 < 23 Subtract 3: 16 ≤ 4x < 20 Divide by 4: 4 ≤ x < 5 So viable numbers in our solution set must be larger or equal to 4 and smaller than 5. Therefore, the only option which does not work here is 5 itself.
• Question 5

Find the solution set for this inequality:

5x + 1 < 3x + 5

x<2
x < 2
x< 2
x <2
EDDIE SAYS
This is the most common type of question you will face. Just remember our rules: 1) Eliminate one of the terms of x. Subtract 3x from both sides: 2x + 1 < 5 2) Get x by itself. Subtract 1 from both sides: 2x < 4 Divide by 2: x < 2 3) Rearrange the inequality if necessary. This is not necessary as our x is already on the left-hand side here.
• Question 6

Find the solution set for this inequality:

3 < 3x < 9

Don't put any spaces in your answer or you may be marked incorrectly.

11 < x < 3
EDDIE SAYS
As this is a three-parter, we have to do the same thing to all three parts of this inequality. To isolate x, we need to divide by 3: 3 ÷ 3 < 3x ÷ 3 < 9 ÷ 3 1 < x < 3 Did you remember to type your answer in without spaces?
• Question 7

Match each inequality below with its solution set.

## Column B

6x + 1 > 2x + 9
x > 2
2x + 2 < 14 - x
x < 4
x + 8 > 7x - 4
x < 2
EDDIE SAYS
Just remember the rules to apply: 1) Eliminate one of the terms in x. 2) Get x by itself by doing the same to both sides. 3) Rearrange the inequality if necessary. Let's take these one at a time. 6x + 1 > 2x + 9 4x + 1 > 9 4x > 8 4x ÷ 4 > 8 ÷ 4 x > 2 2x + 2 < 14 - x 3x + 2 < 14 3x < 12 3x ÷ 3 < 12 ÷ 3 x < 4 x + 8 > 7x - 4 8 > 6x - 4 12 > 6x 12 ÷ 6 > 6x ÷ 6 2 > x As the x is on the left here, we need to switch its sides and flip the sign: x < 2 How did you get on?
• Question 8

Match each inequality below with its solution set.

## Column B

7 < 3x + 1 < 16
2 < x < 5
-7 < 4x - 3 < 9
-1 < x < 3
-9 < 5x + 6 < 1
-3 < x < -1
EDDIE SAYS
How does your working compare to ours below? 7 < 3x + 1 < 16 6 < 3x < 15 6 ÷ 3 < 3x ÷ 3 < 15 ÷ 3 2 < x < 5 -7 < 4x - 3 < 9 -4 < 4x < 12 -4 ÷ 4 < 4x ÷ 4 < 12 ÷ 4 -1 < x < 3 -9 < 5x + 6 < 1 -15 < 5x < -5 -15 ÷ 5 < 5x ÷ 5 < -5 ÷ 5 -3 < x < -1 Don't forget to use a pen and paper to record your working. You will get marks in your exam for your working even if you don't reach the correct answer, so it's really important to get into the habit of recording this accurately.
• Question 9

Find the solution set for this inequality:

4x - 7 > 6x - 13

x < 3
x<3
x <3
x< 3
EDDIE SAYS
Let's follow our three rules. 1) 4x is the smallest term to eliminate: 4x - 7 - 4x > 6x - 4x - 13 -7 > 2x - 13 2) -7 + 13 > 2x - 13 + 13 6 > 2x 6 ÷ 2 > 2x ÷ 2 3 > x 3) x is on the left-hand side, so we need to swap the signs and flip the symbols: 3 > x x < 3 Can you recall these rules by heart yet?
• Question 10

Find the solution set for this inequality:

7 < 4x - 5 < 17

Don't put any spaces in your answer and express any number less than 1 as decimals or you may be marked incorrectly.

33 < x < 5.5
EDDIE SAYS
As it's a three-part inequality, we have to be sure to do the same thing to all three parts of this inequality. Here is our working for this one: 7 < 4x - 5 < 17 7 + 5 < 4x - 5 + 5 < 17 + 5 12 < 4x < 22 12 & divide; 4 < 4x & divide; 4 < 22 & divide; 4 3 < x < 22/4 3 < x < 5.5 Don't worry about the last number being a decimal. This is perfectly valid. Congratulations on completing this activity! If you are ready for a challenge, why don't you give the Level 3 activity on solving inequalities a try?
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