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In this worksheet, students will find solutions to quadratic inequalities by solving them as quadratic equations and interpreting quadratic graphs.

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 quadratic inequalities can seem like a daunting task, but we just need to break these questions down and choose the best approach from our options.

To be able to solve these, you will need to be able to solve quadratic equations (either by factorising or using formula) and sketch quadratic graphs.

Let's look at a couple of examples to see how we can approach these.

e.g. Solve the inequality x2 - 2x - 3 ≥ 0.

Step 1:

Imagine that this is an equation and solve it as normal using either factorisation or the quadratic formula:

x = -1 and x = 3

So if we draw the graph for y = x2 - 2x - 3, these two points are where the graph crosses the x axis.

Step 2:

Step 3:

Identify the parts of the graph that satisfy the inequality: x2 - 2x - 3 ≥ 0

Since we are looking for the points that are greater than or equal to 0, we need to highlight the following sections (shown in red):

We can see that there are two sections highlighted in red.

This means we need two inequalities to describe the solutions to our inequality:

x ≤ -1 and x ≥ 3

e.g. Solve the inequality x2 - x - 20 < 0.

Step 1:

Imagine that this is an equation and solve it as normal using either factorisation or the quadratic formula:

x = -4 and x = 5

So if we draw the graph for y = x2 - x - 20, these two points are where the graph crosses the x axis.

Step 2:

Step 3:

Identify the parts of the graph that satisfy the inequality: x2 - x - 20 < 0

Since we are looking for the points that are less than 0, we need to highlight the following section (shown in red):

We can see that there is only one section highlighted in red.

This means we only need one inequality to describe the solutions to our inequality:

-4 < x < 5

In this activity, we will find solutions to quadratic inequalities by solving them as quadratic equations and interpreting quadratic graphs.

What are the two possible solutions to the quadratic equation x2 + x - 12 = 0?

What are the two possible solutions to the quadratic equation x2 + 3x + 2 = 0?

Consider this equation:

x2 + bx + c ≤ 0

Would the solution to this equation be above or below the x axis?

Above

Below

x<sup>2</sup> + bx + c > 0

How many inequalities would we need to use to describe the solutions?

Which of the following two diagrams illustrate the solutions to the quadratic equation below in red?

x2 + bx+ c ≥ 0

Which of the following is a viable solution for x<sup>2</sup> - x - 12 ≥ 0?

x ≤ 3

x < -3

x > -3

x < 4

x ≥ 4

x > 4

-3 ≤ x ≤ 4

What are the solutions for x2 - 1 < 0?

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

What are the solutions for x2 - 7x + 12 < 0?

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

What are the solutions for x2 - 1 > 0?

In the first space you should use either a < or > symbol, and in the second a number.

What are the solutions for x2 - 7x + 12 > 0?

In the first space you should use either a < or > symbol, and in the second a number.

• Question 1

What are the two possible solutions to the quadratic equation x2 + x - 12 = 0?

EDDIE SAYS
We wanted to start off with a quadratic equation so you can practise your skills here before we move on to inequalities. We can solve this equation using factorisation: (x - 3)(x + 4) = 0 In order to reach an answer of 0, we can assume that one of the brackets must calculate to reach 0. Therefore, x can be +3 (as 3 - 3 = 0) or -4 (as -4 + 4 = 0). Let's try one more equation before we move on to tackle inequalities.
• Question 2

What are the two possible solutions to the quadratic equation x2 + 3x + 2 = 0?

EDDIE SAYS
Here's one more equation to ease you in before the inequalities. This one factorises to: (x + 1)(x + 2) = 0 As one of these brackets must equal 0, we know that x must be -1 (as -1 + 1 = 0) or -2 (as -2 + 2 = 0). Are you ready for some quadratic inequalities now?
• Question 3

Consider this equation:

x2 + bx + c ≤ 0

Would the solution to this equation be above or below the x axis?

Below
EDDIE SAYS
Our inequality total is ≤ 0, which means that our solutions must be below the axis Remember that when an inequality total is < or ≤ 0 this means we will only need one inequality to describe the solutions. Whereas > or ≥ 0 means there will be two inequalities to describe the solutions. Refer back to the graphs in our Introduction to see these two varieties in action.
• Question 4

x<sup>2</sup> + bx + c > 0

How many inequalities would we need to use to describe the solutions?

2
EDDIE SAYS
This question is really asking us to consider whether these solutions on this graph will be above or below the x axis. Our inequality is > 0 so our solutions must be above the axis. This means we need two inequalities to describe the solutions here, as they are within two distinct sections of our graph. Does that make sense?
• Question 5

Which of the following two diagrams illustrate the solutions to the quadratic equation below in red?

x2 + bx+ c ≥ 0

EDDIE SAYS
As this inequality uses a 'greater than or equal to' sign, the section that satisfies this will be above the x axis.
• Question 6

Which of the following is a viable solution for x<sup>2</sup> - x - 12 ≥ 0?

x ≤ 3
x ≥ 4
EDDIE SAYS
This is easiest to work out using a process of elimination. This inequality is greater than zero, which tells us two things: 1) Two inequalities are needed to describe this; 2) It must include the sign ≥ or ≤. If we factorise x2 - x - 12 = 0 as an equation, we reach: (x - 3)(x + 4) = 0 As one of these brackets must equal 0, then x = 3 or x = -4.
• Question 7

What are the solutions for x2 - 1 < 0?

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

-1-1 < x < 1
EDDIE SAYS
If we solve this as an equation, we find that x = -1 and x = 1. As the inequality equals less than 0, the solutions here will be underneath the axis so we will only need one solution. We need to maintain the symbol from our starting inequality and insert our values for x to reach: -1 < x < 1
• Question 8

What are the solutions for x2 - 7x + 12 < 0?

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

33 < x < 4
EDDIE SAYS
If we solve this as an equation, we find that: x = 3 and x = 4 As the inequality uses a 'less than' sign, the solutions will be underneath the axis so we will only need one, combined solution. So put our values for x together with the same inequality symbol to find our answer. Just one more question to go!
• Question 9

What are the solutions for x2 - 1 > 0?

In the first space you should use either a < or > symbol, and in the second a number.

EDDIE SAYS
If we solve this as an equation, we find that: x = -1 and x = 1 As the inequality uses a 'greater than' symbol, the solutions will be above the axis and we will need two separate inequalities.
• Question 10

What are the solutions for x2 - 7x + 12 > 0?

In the first space you should use either a < or > symbol, and in the second a number.

EDDIE SAYS
If we solve this as an equation, we find that: x = 3 and x = 4 As the inequality uses a 'greater than' symbol, the solutions will be above the axis and we will need two separate inequalities. Great work completing this activity, as this was a top-level concept to master.
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