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Solve Quadratic Inequalities

In this worksheet, students will practise solving quadratic inequalities.

'Solve Quadratic 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

Solving quadratic inequalities can be a bit daunting unless you think logically about it.

To be able to solve these, you will need to be able to solve quadratic equations ( It doesn't matter if you factorise or use the formula)  and sketch quadratic graphs.

 

Let's look at a couple of examples.

 

Example 1: Solve the inequality x2 - 2x - 3 ≥ 0

Step 1: Imagine that this is an equation and solve it (use either factorisation or the formula)

x = -1 and x = 3

You should know that if we draw the graph for y = x2 - 2x - 3. These two points are where the graph crosses the x axis.

Step 2: Sketch the quadratic graph

Step 3: Identify the parts of the graph that satisfy the inequality.

We have 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.

We can see that there are two sections highlighted in red. This means we need two inequalities to describe the solutions.

x ≤ -1 and x ≥ 3

 

Example 2: Solve the inequality x2 - x - 20 < 0

Step 1: Imagine that this is an equation and solve it (use either factorisation or the formula)

x = -4 and x = 5

You should know that if we draw the graph for y = x2 - x - 20. These two points are where the graph crosses the x axis.

Step 2: Sketch the quadratic graph

Step 3: Identify the parts of the graph that satisfy the inequality.

We have 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.

We can see that there is only one section highlighted in red. This means we only need one inequality t describe the solutions.

-4 < x < 5

 

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

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

For the inequality x2 + bx + c ≤ 0. Would the solution be above or below the x axis?

above

below

For the inequality x2 + bx + c >0 0. How many inequalities would you need to use to describe the solutions?

Which of the following two diagrams illustrate the solutions for x2 + bx+ c ≥ 0



Which of the following is the solution for x2 -x - 12 ≥ 0?

x ≤ 3

x < -3

x > -3

x < 4

x ≥ 4

x > 4

-3 &le' s ≤ 4

What are the solutions for x2 - 1 < 0

 

(Don't put any spaces in your answer)

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

 

(Don't put any spaces in your answer)

What are the solutions for x2 - 1 > 0

 

(Don't put any spaces in your answer)

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

 

(Don't put any spaces in your answer)

  • Question 1

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

CORRECT ANSWER
EDDIE SAYS
A nice easy one to start with. This one does factorise to (x-3)(x+4)=0 From this we can get our solutions.
  • Question 2

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

CORRECT ANSWER
EDDIE SAYS
A nice easy one to start with. This one does factorise to (x+1)(x+2)=0 From this we can get our solutions.
  • Question 3

For the inequality x2 + bx + c ≤ 0. Would the solution be above or below the x axis?

CORRECT ANSWER
below
EDDIE SAYS
Our inequality is ≤ 0 so our solutions must be below the axis Remember that this means you only need one inequality to describe the solutions.
  • Question 4

For the inequality x2 + bx + c >0 0. How many inequalities would you need to use to describe the solutions?

CORRECT ANSWER
2
EDDIE SAYS
Our inequality is > 0 so our solutions must be above the axis Remember that this means you need two inequalities to describe the solutions as they are two distinct sections.
  • Question 5

Which of the following two diagrams illustrate the solutions for x2 + bx+ c ≥ 0

CORRECT ANSWER

EDDIE SAYS
As the inequality has the greater than or equal sign, the section that satisfies is above the axis.
  • Question 6

Which of the following is the solution for x2 -x - 12 ≥ 0?

CORRECT ANSWER
x ≤ 3
x ≥ 4
EDDIE SAYS
This is a bit of a process of elimination. The inequality is greater than zero, this tells us two things. 1) Two inequalities are needed to describe this 2) It must include either ≥ or ≤
  • Question 7

What are the solutions for x2 - 1 < 0

 

(Don't put any spaces in your answer)

CORRECT ANSWER
-1
EDDIE SAYS
If we solve this, we get x = -1 and x = 1. As the inequality is a less than, the solutions are underneath the axis and will only need one solution.
  • Question 8

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

 

(Don't put any spaces in your answer)

CORRECT ANSWER
3
EDDIE SAYS
If we solve this, we get x = 3 and x = 4. As the inequality is a less than, the solutions are underneath the axis and will only need one solution.
  • Question 9

What are the solutions for x2 - 1 > 0

 

(Don't put any spaces in your answer)

CORRECT ANSWER
EDDIE SAYS
If we solve this, we get x = -1 and x = 1. As the inequality is a a more than, the solutions area bove the axis and will need two inequalities.
  • Question 10

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

 

(Don't put any spaces in your answer)

CORRECT ANSWER
EDDIE SAYS
If we solve this, we get x = 3 and x = 4 As the inequality is a a more than, the solutions area bove the axis and will need two inequalities.
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