9

Quadratic Inequality in
One Variable
Learner's Module in Mathematics 9
Quarter 1 ● Module 5

VIC JOMAR M. LADERAS

Developer

Department of Education • Cordillera Administrative Region

NAME: ________________________ GRADE AND SECTION: _______________

TEACHER: ___________________________________SCORE: ______________
 Republic of the Philippines
DEPARTMENT OF EDUCATION
Cordillera Administrative Region
SCHOOLS DIVISION OF BAGUIO CITY
Military Cut-off, Baguio City

Published by:
DepEd Schools Division of Baguio City
Curriculum Implementation Division

COPYRIGHT NOTICE
2020

Section 9 of Presidential Decree No. 49 provides:

“No copyright shall subsist in any work of the Government of the Philippines.
However, prior approval of the government agency of office wherein the work is
created shall be necessary for exploitation of such work for profit.”

This material has been developed for the implementation of K-12 Curriculum
through the DepEd Schools Division of Baguio City – Curriculum Implementation
Division (CID). It can be reproduced for educational purposes and the source must be
acknowledged. Derivatives of the work including creating an edited version, an
enhancement or a supplementary work are permitted provided all original work is
acknowledged and the copyright is attributed. No work may be derived from this
material for commercial purposes and profit.

ii
 PREFACE

This module is a project of the DepEd Schools Division of Baguio City through
the Curriculum Implementation Division (CID) which is in response to the
implementation of the K to 12 Curriculum.

This Learning Material is a property of the Department of Education, Schools

Division of Baguio City. It aims to improve students’ academic performance specifically
in Mathematics.

Date of Development : June 2020

Resource Location : DepEd Schools Division of Baguio City
Learning Area : Mathematics
Grade Level :9
Learning Resource Type : Module
Language : English
Quarter/Week : Q1/W5
Learning Competency/Code : Illustrates quadratic inequalities in
one variable (M9AL-If-1);
solves quadratic inequalities in
one variable (M9AL-If-g-1)

iii
 ACKNOWLEDGEMENT

The developer would like to express his gratitude to those who, in one way or
another, have contributed in the development of this learning material.

Appreciation for all the collaboration and cooperation given by the Grade-9
Mathematics teachers. Boundless gratitude goes to his friends for sharing their time
and talent in crafting this learning resource and to all the students of Baguio City
National High School who are hoping to learn despite this pandemic. Lastly, thanks to
their school’s supervisory office led by their school principal, Madam Brenda M. Cariño
and the DepEd Division of Baguio City for all the support.

Development Team
Author: Vic Jomar M. Laderas
Illustrators: Ian T. Tomin (Cover Art)
Vic Jomar M. Laderas

School Learning Resources Management Committee

Brenda M. Cariño School Principal
Editha L. Laop Subject/ Learning Area Specialist
Niño E. Martinez Subject/ Learning Area Specialist
Sherwin Fernando School LR Coordinator

Quality Assurance Team

Francisco C. Copsiyan EPS – Mathematics
Lourdes B. Lomas-e PSDS – BCNHS District

Learning Resource Management Section Staff

Loida C. Mangangey EPS – LRMDS
Victor A. Fernandez Education Program Specialist II - LRMDS
Christopher David G. Oliva Project Development Officer II – LRMDS
Priscilla A. Dis-iw Librarian II
Lily B. Mabalot Librarian I

CONSULTANTS

JULIET C. SANNAD, EdD

Chief Education Supervisor – CID

SORAYA T. FACULO, PhD

Asst. Schools Division Superintendent

MARIE CAROLYN B. VERANO, CESO V

Schools Division Superintendent

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 TABLE OF CONTENTS

Page
COPYRIGHT NOTICE…………………………………………….……...….…. ii
PREFACE ………………………………………………………………………... iii
ACKNOWLEDGEMENT………………………………………………………… iv
TABLE OF CONTENTS……………………………………………….……....... v
TITLE PAGE ………………………………………………………………...…… 1
What I Need to Know ……………………………………………………..…….. 2
What I Know………………………..……………………………….……………. 3
What’s In………………………………………………………………….………. 5
Activity 1. Don’t Forget Me!
What’s New ………………..……………….……………………...……..……… 6
Activity 2. Do you Know Me?
What Is It…….…………………………………………………………….……… 6
What’s More.....…….………………………………………..…...……………… 11
Activity 3. You Can Do It Yourself!
What I Have Learned.................................................................................... 12
Activity 4. I Can Tell You What I Have Learned!
What I Can Do.............................................................................................. 12
Activity 5. Fence My Garden!
Post Assessment …………...…………………………………………….…… 14
Additional Activity ……………………………………………………………….. 16
Activity 6. Extend Your Understanding!
Answer Key…………………………………………………………………….…. 17
Reference Sheet…………………………………………………...…….…….… 20

v
Quadratic Inequality in
One Variable
Learner's Module in Mathematics 9
Quarter 1 ● Module 5

VIC JOMAR M. LADERAS

Developer

Department of Education • Cordillera Administrative Region

 What I Need to Know

Hello learner! This module was designed and written with you in mind. Primarily,
its scope is to develop your understanding on the relationship between the roots and
coefficients of a quadratic equation.

While going through this module, you are expected to:

1. illustrate quadratic inequalities;
2. solve quadratic inequalities in one variable; and
3. graph quadratic inequalities in one variable.
By the way, always remember to use the answer sheet for you to write
your answers on the different activities and assessments presented in this
learning module. DO NOT ANSWER HERE directly.
Now, here is an outline of the different parts of your learning module. The
descriptions will guide you on what to expect on each part of the module.

Icon Label Description

What I Need to Know This states the learning objectives that you need to
achieve as you study this module.

What I Know This is to check what you already know about the lesson
on this module. If you answered all the questions here
correctly, then you may skip studying this module.

What’s In This connects the current lesson with a topic or concept

necessary to your understanding.

What’s New This introduces the lesson to be tackled through an

activity.

What Is It This contains a brief discussion of the learning module

lesson. Think of it as the lecture section of the lesson.

What’s More These are activities to check your understanding and to

apply what you have learned from the lesson.

What I have Learned This generalizes the essential ideas tackled from this
module.
What I Can Do This is a real-life application of what you have learned.

Post Assessment This is an evaluation of what you have learned from this
learning material.

Additional Activity This is an activity that will strengthen and fortify your
knowledge about the lesson.

2
 What I Know

If you answer all the test items correctly in this pre-assessment, then you may
skip studying this learning material and proceed to the next learning module.

DIRECTION: Let us determine how much you already know about quadratic
inequalities. Read and understand each item, then choose the letter of your answer
and write it on your answer sheet.

1) How many solutions does a linear inequality in one variable have?

A. one C. three
B. two D. infinite

2) Solve: 2𝑥 < −6
A. 𝑥 < −3 C. 𝑥 > −3
B. 𝑥 < 3 D. 𝑥 > 3
For numbers 3 – 4, refer to the graph below.

3) Which of the following is not a solution to the inequality?

A. 3 C. 0
B. 2 D. −3

4) Which inequality describes the graph?

A. 𝑥 < 2 C. 𝑥 > 2
B. 𝑥 ≤ 2 D. 𝑥 ≥ 2

5) How many solutions does a quadratic inequality in one variable have?

A. one C. three
B. two D. infinite

6) ____________is a value of 𝑥 for which an inequality equals 0 or is undefined.

A. critical number C. domain
B. solution D. range

7) 𝑥 > −3 ∩ 𝑥 < −2 is the same as ________________.

A. { } 𝑜𝑟 ∅ C. −3 < 𝑥 < −2
B. −2 < 𝑥 < −3 D. −3 > 𝑥 > −2

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8) In a set, the word “and” is symbolized as ________.
A. ∪ C. ∅
B. ∩ D. ≠

9) In 3 ≤ 𝑥 < 7,which is not a solution?

A. 7 C. 4
B. 3 D. 5

10) Find the solution of 𝑥 2 + 6𝑥 + 8 ≤ 0.

A. −4 ≥ 𝑥 ≥ −2 C. 4 ≥ 𝑥 ≥ 2
B. −4 ≤ 𝑥 ≤ −2 D. 4 ≤ 𝑥 ≤ 2

11) What is the solution of 𝑥 2 + 𝑥 − 12 > 0.

A. −4 < 𝑥 < 3 C. 𝑥 > −4 ∪ 𝑥 < 3
B. −4 > 𝑥 > 3 D.𝑥 < −4 ∪ 𝑥 > 3
For numbers 12 – 13, refer to the graph below.

12) What are the critical numbers?

A. −5 𝑎𝑛𝑑 − 3 C. −4.5, −4.0, 𝑎𝑛𝑑 − 3.5
B. 5 𝑎𝑛𝑑 3 D. none of the above

13) What is the solution of the inequality?

A. 5 > 𝑥 > 3 C. 5 < 𝑥 < 3
B. −5 > 𝑥 > −3 D. −5 < 𝑥 < −3

For numbers 14 – 15, refer to the graph below.

14) The numbers −2 𝑎𝑛𝑑 3 are solutions of the inequality.

A. true C. sometimes true
B. false D. cannot be determined

15) What is the solution of the inequality?

A. 𝑥 < −2 ∪ 𝑥 > 3 C. 𝑥 > −2 ∪ 𝑥 < 3
B. 𝑥 < −2 ∩ 𝑥 > 3 D. 𝑥 > −2 ∩ 𝑥 < 3

4
 What’s In

Before you proceed to the next lesson, it is very

important that you take a simple recall on finding the
roots of quadratic equation; and solving and graphing
linear inequality in one variable.

Activity 1. Don’t Forget Me!

A. Find My Roots! Find the roots of each equation using any method.

Roots
Equation
𝑟1 𝑟2
1. (𝑥 + 5)(𝑥 − 3) = 0
2. 𝑥 2 + 7𝑥 + 10 = 0
3. 𝑥 2 − 2𝑥 − 24 = 0
4. 𝑥 2 − 5𝑥 + 6 = 0
5. 2𝑥 2 + 5𝑥 + 2 = 0

B. Solve and Graph Me! Find the solution and graph each linear inequality on the
number line.
ex. REMEMBER: In graphing,
2𝑥 − 4 > 6 (open circle) is to be used
when the inequality symbol is
2𝑥 > 10
< 𝑜𝑟 >, which means that
𝑥>5 the critical number is a
solution while (closed
circle) is to be used when the
inequality symbol is ≤ 𝑜𝑟 ≥,
which means that the critical
number is not a solution.

1. 𝑥 − 2 > 10
2. 𝑥 + 6 ≤ −2
3. 2𝑥 < 6
4. 4𝑥 − 6 < 2𝑥 − 2
5. 4𝑥 ≥ 2(𝑥 − 8)

5
 What’s New

Activity 2. Do You Know Me?

Categorize each into quadratic inequality or not quadratic inequality in one
variable.

1. 𝑥 2 + 3𝑥 − 2 > 0 6. 4𝑥 ≠ 20
2. 4𝑥 2 + 7𝑥 + 1 = 0 7. 7𝑥 2 + 5𝑥 > 2𝑥 − 2
3. 2𝑥 − 5 ≤ 0 8. 4𝑥 2 + 5 < 2𝑥(3𝑥 − 2)
2
4. 4𝑥 + 10 = 3𝑥 9. > 3𝑥
𝑥
5. 2𝑥 2 ≥ 50 10. (𝑥 + 2)(7 − 3𝑥) < 2

What Is It

You have just reviewed how to find the roots of a quadratic

equation, and how to solve and graph linear inequality in one
variable. This time let’s study how these would help you find
the solution of a quadratic inequality in one variable. There
are different ways of expressing solutions to these inequalities
but in our lesson, you will only be taught how to express
solution in inequality notation. Other notations are set and
interval notations.

A quadratic inequality in one variable is an inequality that contains a

polynomial degree 2 and can be written in any of the following forms:
𝑎𝑥 2 + 𝑏𝑥 + 𝑐 > 0 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 ≥ 0

𝑎𝑥 2 + 𝑏𝑥 + 𝑐 < 0 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 ≤ 0

where 𝑎, 𝑏, 𝑎𝑛𝑑 𝑐 are real numbers and 𝑎 ≠ 0.

Examples: 1. 2𝑥 2 + 7𝑥 − 3 > 0
2. 4𝑥 2 ≤ 9 4𝑥 2 − 9 ≤ 0
3. (2𝑥 + 1)(3𝑥 − 1) < 0 2𝑥 2 + 𝑥 − 1 < 0

6
 Unlike quadratic equation which has exactly two solutions, quadratic
inequality in one variable has infinite solutions. There are two ways in solving
quadratic inequality – algebraic method and graphical method.

A. ALGEBRAIC METHOD

In solving quadratic inequalities using algebraic method, we need to

consider two basic theorems, such as:

Theorem 1: The product of two quantities is positive if both are

positive or both are negative quantities.

a𝑏 > 0, 𝑖𝑓 𝑎 > 0 𝑎𝑛𝑑 𝑏 > 0, 𝑜𝑟 𝑎 < 0 𝑎𝑛𝑑 𝑏 < 0

If 𝑎𝑏 > 0, then the product is positive.
(+)(+) = + 𝑜𝑟 (−)(−) = +

Theorem 2: The product of two quantities is negative if one quantity

is positive and the other quantity is negative.

a𝑏 < 0, 𝑖𝑓 𝑎 > 0 𝑎𝑛𝑑 𝑏 < 0, 𝑜𝑟 𝑎 < 0 𝑎𝑛𝑑 𝑏 > 0

If 𝑎𝑏 < 0, then the product is negative, if
(+)(−) = − 𝑜𝑟 (−)(+) = −

NOTE: Theorems 1 and 2 will be used even if the

inequality symbols are ≤ 𝑜𝑟 ≥.

Steps in Solving Quadratic Inequality Using ALGEBRAIC METHOD

1. Write the quadratic inequality in the form
<
>
≤
≥
𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 𝟎 , where 𝑎, 𝑏, 𝑎𝑛𝑑 𝑐 are integers.
2. Factor the quadratic expression. (If the expression is not factorable, the
graphical method is more appropriate to use.)
3. Use theorems 1 or 2 whichever is applicable in determining the solution.

REMEMBER: In writing solution of quadratic inequality,

“and” is symbolized as ∩ which means
intersection while “or” is symbolized as ∪
which means union.

7
Illustrative example 1: Find the solution of 𝑥 2 + 3𝑥 − 4 < 0.

Illustration Step/Explanation

𝑥 2 + 3𝑥 − 4 < 0 Write the general form.

(𝑥 + 4)(𝑥 − 1) < 0 Factor the expression.
Use theorem 2. The product in factored
Case 1: 𝑥 + 4 < 0 and 𝑥 − 1 > 0
form is less than zero. It means that
or
either of the factors is positive or
Case 2: 𝑥 + 4 > 0 and 𝑥 − 1 < 0
negative.
Case 1: 𝑥 + 4 < 0 and 𝑥 − 1 > 0 Solve each case. In case 1, the
𝑥 < −4 and 𝑥 > 1 intersection of 𝑥 < −4 and 𝑥 > 1 is
∴ ∅ 𝑜𝑟 { } ∅ 𝑜𝑟 { } since no value satisfies both
inequalities. In case 2, the intersection of
Case 2: 𝑥 + 4 > 0 and 𝑥 − 1 < 0 𝑥 > −4 and 𝑥 < 1
𝑥 > −4 and 𝑥 < 1 is 𝑥 > −4 ∩ 𝑥 < 1. It can also be written as
∴ 𝑥 > −4 ∩ 𝑥 < 1 or it can be −4 < 𝑥 < 1.
written as −4 < 𝑥 < 1
Combine solutions of
∴ 𝒙 > −𝟒 ∩ 𝒙 < 𝟏
Cases 1 and 2. Since Case 1 has no
which can be written as
solutions, only solution of Case 2 is
−𝟒 < 𝒙 < 𝟏
considered.
The solution of 𝑥 2 + 3𝑥 − 4 < 0 is 𝑥 > −4 ∩ 𝑥 < 1 . It can also be
written as −4 < 𝑥 < 1.

Illustrative example 2: Find the solution of 𝑥 2 + 3𝑥 ≥ 10.

Illustration Step/Explanation

𝑥 2 + 3𝑥 ≥ 10
Write the general form.
𝑥 2 + 3𝑥 − 10 ≥ 0
(𝑥 + +5)(𝑥 − 2) ≥ 0 Factor the expression.
Use theorem 1. The product in factored
Case 1: 𝑥 + 5 ≥ 0 and 𝑥 − 2 ≥ 0 form is greater than or equal to zero.
or Aside from zero factors, the other
Case 2: 𝑥 + 5 ≤ 0 and 𝑥 − 2 ≤ 0 possible factors are both positive or
both negative.
Case 1: 𝑥 + 5 ≥ 0 and 𝑥 − 2 ≥ 0
𝑥 ≥ −5 and 𝑥 ≥ 2 Solve each case. In case 1, the
∴𝑥≥2 intersection of 𝑥 ≥ −5 and 𝑥 ≥ 2 is 𝑥 ≥
2. In case 2, the intersection of
Case 2: 𝑥 + 5 ≤ 0 and 𝑥 − 2 ≤ 0 𝑥 ≤ −5 and 𝑥 ≤ 2 is 𝑥 ≤ −5.
𝑥 ≤ −5 and 𝑥 ≤ 2
∴ 𝑥 ≤ −5

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 ∴ 𝒙 ≤ −𝟓 𝒐𝒓 𝒙 ≥ 2
Combine solutions of
which can be written as
Cases 1 and 2.
𝒙 ≤ −𝟓 ∪ 𝒙 ≥ 2
The solution of 𝑥 2 + 3𝑥 ≥ 10 is 𝑥 ≤ −𝟓 ∪ 𝒙 ≥ 2.

How did you find the algebraic method? If you are having
difficulty in understanding the method, surely, you will
find the graphical method easier to learn. We will be using
the same problems so that we can check our answers.
Don’t worry I will make it easy for you to understand.
Have fun!

B. GRAPHICAL METHOD

Steps in Solving Quadratic Inequality Using GRAPHICAL METHOD

1. Determine the critical numbers (values of 𝑥 for which an inequality
equals 0 or is undefined). It is the same as finding the roots of a
quadratic equation.
2. Plot the critical numbers on the number line. Since there two always two
critical numbers, the number line will be divided into three intervals.
3. To determine solutions, select arbitrary numbers from each interval and
test them in the inequality.
4. Graph solutions on the number line by making broad straight lines, then
write the inequality notation of the solution.

Illustrative example 1: Graph and find the solution of 𝑥 2 + 3𝑥 − 4 < 0.

1. Change inequality symbol to equal sign, then solve for the roots of the
quadratic equation 𝑥 2 + 3𝑥 − 4 = 0. It can be solved by factoring.
𝑥 2 + 3𝑥 − 4 < 0

𝑥 2 + 3𝑥 − 4 = 0
(𝑥 + 4)(𝑥 − 1) = 0
𝑥+4=0 𝑥−1=0
𝑥 = −4 𝑥=1
2. Plot the critical numbers −4 𝑎𝑛𝑑 1 on the number line. Use open circle
since the inequality symbol is < . It means that −4 𝑎𝑛𝑑 1 are not parts
of the solution.

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 3. Select −5, 0 𝑎𝑛𝑑 2 from each interval, then test in the inequality.

𝑥 2 + 3𝑥 − 4 < 0 𝑥 2 + 3𝑥 − 4 < 0 𝑥 2 + 3𝑥 − 4 < 0

? ? ?
(−5)2 + 3(−5) − 4 < 0 (0)2 + 3(0) − 4 < 0 (2)2 + 3(2) − 4 < 0
6≮0 −4 < 0 6≮0
not solution solution not solution

4. Connect the two open circles to show the solution of the inequality.

−𝟒 < 𝒙 < 𝟏

The solution of 𝑥 2 + 3𝑥 − 4 < 0 is −4 < 𝑥 < 1.

Illustrative example 2: Graph and find the solution of 𝑥 2 + 3𝑥 ≥ 10 .

1. Change inequality symbol to equal sign, then solve for the roots of the
quadratic equation 𝑥 2 + 3𝑥 = 10. It can be solved again by factoring.
𝑥 2 + 3𝑥 ≥ 10

𝑥 2 + 3𝑥 = 10
𝑥 2 + 3𝑥 − 10 = 0
(𝑥 + 5)(𝑥 − 2) = 0
𝑥+5=0 𝑥−2=0
𝑥 = −5 𝑥=2

2. Plot the critical numbers −5 𝑎𝑛𝑑 2 on the number line. Use closed circle
since the inequality symbol is ≥ . It means that −5 𝑎𝑛𝑑 2 are parts of
the solution.

10
 3. Select −6, 0 𝑎𝑛𝑑 3 from each interval, then test in the inequality.

𝑥 2 + 3𝑥 ≥ 10 𝑥 2 + 3𝑥 ≥ 10 𝑥 2 + 3𝑥 ≥ 10
? ? ?
(−6)2 + 3(−6) ≥ 10 (0)2 + 3(0) ≥ 10 (3)2 + 3(3) ≥ 10
18 ≥ 10 0 ≱ 10 18 ≥ 10
solution not solution solution

4. Make arrows on both sides of the number line to show the solution of
the inequality.
𝒙 ≤ −𝟓 𝒙≥𝟐

The solution of 𝑥 2 + 3𝑥 ≥ 10 is 𝑥 ≤ −5 𝑜𝑟 𝑥 ≥ 2. It can also be

written as 𝑥 ≤ −5 ∪ 𝑥 ≥ 2.

REMINDER: When the quadratic expression is ≤ 𝑜𝑟 < than zero, the solution is
always the middle interval, whereas when the quadratic expression is
≥ 𝑜𝑟 > than zero, the solutions are the two intervals on opposite sides
of the number line.

What’s More
Activity 3: You Can Do It Yourself!
A. Graph and find the solution of each quadratic inequality. Show
complete solution.
1. (𝑥 − 3)(𝑥 + 5) > 0
2. 𝑥 2 + 5𝑥 + 4 < 0
3. 𝑥 2 + 4𝑥 ≥ 12

B. Using the algebraic method, find the solution of each.

1. (𝑥 − 5)(𝑥 + 4) ≤ 0
2. 2𝑥 2 − 5𝑥 − 3 > 0

11
 What I Have Learned
Activity 4: I Can Tell You What I Have Learned
Fill in the blanks. (For this part, you are no longer allowed to refer to the
previous discussion when answering.)

1. A linear inequality in one variable has ____________ solutions.

2. 2𝑥 2 + 5𝑥 − 7 > 0 is an example of ______________ inequality in one
variable.
3. A quadratic inequality in one variable has a degree ______.
4. A quadratic inequality in one variable has _________ solutions.
5. ___________________ is a value of x for which an inequality equals 0 or is
undefined.
6. In inequality, use _______ circle when the symbol is < 𝑜𝑟 >.
7. In inequality, use _______ circle when the symbol is ≤/𝑜𝑟 ≥.
8. The word “and” is symbolized as ________.
9. The word “or” is symbolized as _________.
10. When the quadratic expression is ≤ 𝑜𝑟 < than zero, the solution is the
__________ interval of the number line.

What I Can Do

Activity 5: Fence My Garden!

Illustrative example:
Victorio has a 40 − 𝑓𝑡 metal fencing material to fence three sides of a
rectangular garden. A tall wooden fence serves as the length of the garden
which is the fourth side. What measures for the width will give an area of at
least 150𝑓𝑡 2 ? wooden fence 40 𝑓𝑡
Let 𝑥 – the width of the garden
Since width is 𝑥 and the metal fencing
𝑥 𝑥
material is 40𝑓𝑡 long, we can represent the
length to be 40 − 2𝑥. 40 − 2𝑥

12
 Illustration Step/Explanation
Remember that in a rectangle,
𝐴 = (40 − 2𝑥)(𝑥)
𝐴𝑟𝑒𝑎 = 𝑙𝑒𝑛𝑔𝑡ℎ 𝑥 𝑤𝑖𝑑𝑡ℎ
𝐴 = 40𝑥 − 2𝑥 2 𝑜𝑟 Simplify the right side by applying
𝐴 = −2𝑥 2 + 40𝑥 distributive property for multiplication
Since the area is at least 150 𝑓𝑡 2 , we
150 ≤ −2𝑥 2 + 40𝑥
have
150 ≤ −2𝑥 2 + 40𝑥 Rewrite the inequality.
2𝑥 2 − 40𝑥 + 150 ≤ 0

𝑥 2 − 20𝑥 + 75 ≤ 0 Simplify the coefficients by dividing

both sides by 2.
(𝑥 − 15)(𝑥 − 5) ≤ 0 Factor the right side.

𝑥 − 5 = 0 ; 𝑥 − 15 = 0 Determine critical numbers.

𝑥 = 5 ; 𝑥 = 15

5 ≤ 𝑥 ≤ 15 Since the quadratic expression is ≤

0, the solution is the middle interval.
5 15 You can test this by selecting
numbers form each interval.

The measures of the width of the rectangular garden are at least 5𝑓𝑡
and at most 15𝑓𝑡. In symbols, 5𝑓𝑡 ≤ 𝑥 ≤ 15𝑓𝑡.

PROBLEM:

Pedro has a 44 − 𝑓𝑡 metal fencing material to fence three sides of a

rectangular garden. A tall wooden fence serves as the length of the garden
which is the fourth side. What measures for the width will give an area of at
least 240𝑓𝑡 2 ?

13
 Post Assessment
DIRECTION: Let us determine how much you have learned from this module. Read
and understand each item, then choose the letter of your answer and write it on your
answer sheet.

1) How many solutions does a linear inequality in one variable have?

A. infinite C. two
B. three D. one

2) Solve: 2𝑥 − 3 < 5
A. 𝑥 < 4 C. 𝑥 > 4
B. 𝑥 < 1 D. 𝑥 > 1
For numbers 3 – 4, refer to the graph below.

3) Which of the following is a solution to the inequality?

A. −4 C. 4
B. 0 D. 8

4) Which inequality describes the graph?

A. 𝑥 < 4 C. 𝑥 > 4
B. 𝑥 ≤ 4 D. 𝑥 ≥ 4

5) How many solutions does a quadratic inequality in one variable have?

A. infinite C. two
B. three D. one

6) ______________ is a value of 𝑥 for which an inequality equals 0 or is undefined.

A. domain C. solution
B. range D. critical number

7) 𝑥 > 3 ∩ 𝑥 < 6 is the same as ________________.

A. { } 𝑜𝑟 ∅ C. 3 > 𝑥 > 6
B. 3 < 𝑥 < 6 D. 3 < 𝑥 > 6

8) In a set, the word “or” is symbolized as ________.

A. ∪ C. ∅
B. ∩ D. ≠

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9) In 1 ≤ 𝑥 < 10,which is not a solution?
A. 1 C. 8
B. 5 D. 10

10) Find the solution of 𝑥 2 − 2𝑥 − 3 < 0.

A. −1 < 𝑥 < 3 C. 1 < 𝑥 < −3
B. −1 > 𝑥 > 3 D. 1 > 𝑥 > −3

11) What is the solution of 𝑥 2 − 6𝑥 + 8 > 0.

A. 2 < 𝑥 < 4 C. 𝑥 < 2 ∪ 𝑥 > 4
B. 2 > 𝑥 > 4 D.𝑥 < −2 ∪ 𝑥 > −4

For numbers 12 – 13, refer to the graph below.

12) What are the critical numbers?

A. 1 𝑎𝑛𝑑 7 C. 2, 4, 𝑎𝑛𝑑 6
B. 0 𝑎𝑛𝑑 8 D. none of the above

13) What is the solution of the inequality?

A. 1 < 𝑥 < 7 C. 0 < 𝑥 < 8
B. 1 ≤ 𝑥 ≤ 7 D. 0 ≤ 𝑥 ≤ 8

For numbers 14 – 15, refer to the graph below.

14) The numbers 3 𝑎𝑛𝑑 5 are solutions of the inequality.

A. true C. sometimes true
B. false D. cannot be determined

15) What is the solution of the inequality?

A. 𝑥 < 3 ∪ 𝑥 > 5 C. 𝑥 ≤ 3 ∪ 𝑥 ≥ 5
B. 𝑥 < 3 ∩ 𝑥 > 5 D. 𝑥 ≤ 3 ∩ 𝑥 ≥ 5

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 Additional Activity
Activity 6: I. Strengthen Your Understanding!

A. Graph and find the solution of each quadratic inequality. Show complete
solution.
1. (𝑥 + 1)(𝑥 + 8) < 0
2. 𝑥 2 − 10𝑥 + 24 > 0
3. 𝑥 2 − 𝑥 ≤ 30

B. Using the algebraic method, find the solution of each.

1. (𝑥 + 7)(𝑥 − 2) > 0
2. 2𝑥 2 − 𝑥 − 3 < 0

II. Challenge Yourself! (Bonus Problem)

𝑥+3
Graph and find the solution of < 0. (Hint: Rational Inequality)
𝑥−2

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 ANSWER KEY
What I Know
1. D 6. A 11. D
2. A 7. C 12. A
3. A 8. B 13. D
4. B 9. A 14. B
5. D 10. B 15. A

Activity 1. Don’t Forget Me!

A. Find My Roots!
Roots
Equation
𝑟1 𝑟2
1 −𝟓 𝟑
2 −𝟓 −𝟐
3 𝟔 −𝟒
4 𝟑 𝟐
𝟏
5 −𝟐 −
𝟐

B. Solve and Graph Me!

1. 𝑥 > 12

2. 𝑥 ≤ −8

3. 𝑥 < 3

4. 𝑥 < 2

5. 𝑥 ≥ −8

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Activity 2. Do You Know Me?
Quadratic Inequality Not Quadratic Inequality
2
𝑥 + 3𝑥 − 2 > 0 4𝑥 2 + 7𝑥 + 1 = 0
2𝑥 2 ≥ 50 2𝑥 − 5 ≤ 0
7𝑥 2 + 5𝑥 > 2𝑥 − 2 4𝑥 + 10 = 3𝑥
4𝑥 2 + 5 < 2𝑥(3𝑥 − 2) 4𝑥 ≠ 20
2
(𝑥 + 2)(7 − 3𝑥) < 2 > 3𝑥
𝑥

Activity 3: You Can Do It yourself!

A. 1.
𝑥 < −5 ∪ 𝑥 > 3

2.
−4 < 𝑥 < −1

3.
𝑥 ≤ −6 ∪ 𝑥 ≥ 2

B. 1. −4 ≤ 𝑥 ≤ 5
1
2. 𝑥 < − ∪ 𝑥 > 3
2

Activity 4: I Can Tell You What I Have Learned

1. infinite 6. open
2. quadratic 7. closed
3. 2 8. ∩
4. Infinite 9. ∪
5. critical number 10. Middle
6.

Activity 5: Fence My Garden!

The measures of the width of the rectangular garden are at least
10𝑓𝑡 and at most 12𝑓𝑡. In symbols, 10𝑓𝑡 ≤ 𝑥 ≤ 12𝑓𝑡.

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Activity 6: Strengthen Your Understanding

A. 1.
−8 < 𝑥 < −1

2.
𝑥 < 4∪𝑥 >6

3.
−5 ≤ 𝑥 ≤ 6

B. 1. 𝑥 < −7 ∪ 𝑥 > 2
3
2. −1 < 𝑥 < 2

BONUS Problem

C.
−3 < 𝑥 < 2

Post Assessment

1. A 6. D 11. C
2. A 7. B 12. A
3. D 8. A 13. A
4. C 9. D 14. A
5. A 10. A 15. C

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REFERENCES

Oronce, Orlando A., and Marilyn O. Mendoza. Exploring Mathematics Intermediate

Algebra. Sampaloc City: REX Book Store, Inc., 2003.

https://www.mathway.com/ProblemWidget.aspx?subject=Algebra&affiliateid=affil1809
2

https://www.desmos.com/calculator/dezoto9tsm

https://www.bitmoji.com

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