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Mathematics
Quarter 1 – Module 5A:
Quadratic Inequalities
9
Mathematics – Grade 9 Self-Learning Module (SLM) Quarter 1 – Module 5A: Quadratic Inequalities First Edition, 2020 Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them.
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Development Team of the Module
Writers: Romelyn E. Salut
Editors: Noel A. Wamar; Rosselle L. Rivac, Feby D. Atay
Reviewers: Ronela S. Molina
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9
Mathematics 9 Quarter 1 – Module 5A:
Quadratic Inequalities
Introductory Message
For the facilitator:
Welcome to the Mathematics 9 Self-Learning Module (SLM) on Quadratic
Inequalities!
This module was collaboratively designed, developed and reviewed by educators
both from public and private institutions to assist you, the teacher or facilitator in
helping the learners meet the standards set by the K to 12 Curriculum while
overcoming their personal, social, and economic constraints in schooling.
This learning resource hopes to engage the learners into guided and independent
learning activities at their own pace and time. Furthermore, this also aims to help
learners acquire the needed 21st century skills while taking into consideration
their needs and circumstances.
In addition to the material in the main text, you will also see this box in the body of
the module:
As a facilitator you are expected to orient the learners on how to use this module.
You also need to keep track of the learners' progress while allowing them to
manage their own learning. Furthermore, you are expected to encourage and assist
the learners as they do the tasks included in the module.
Notes to the Teacher
This contains helpful tips or strategies
that will help you in guiding the learners.
For the learner:
Welcome to the Mathematics 9 Self-Learning Module (SLM) on Quadratic
Inequalities!
The hand is one of the most symbolized part of the human body. It is often used to
depict skill, action and purpose. Through our hands we may learn, create and
accomplish. Hence, the hand in this learning resource signifies that you as a
learner is capable and empowered to successfully achieve the relevant
competencies and skills at your own pace and time. Your academic success lies in
your own hands!
This module was designed to provide you with fun and meaningful opportunities
for guided and independent learning at your own pace and time. You will be
enabled to process the contents of the learning resource while being an active
learner.
This module has the following parts and corresponding icons:
What I Need to Know
This will give you an idea of the skills or
competencies you are expected to learn in
the module.
What I Know
This part includes an activity that aims to
check what you already know about the
lesson to take. If you get all the answers
correct (100%), you may decide to skip this
module.
What’s In
This is a brief drill or review to help you link
the current lesson with the previous one.
What’s New
In this portion, the new lesson will be
introduced to you in various ways such as a
story, a song, a poem, a problem opener, an
activity or a situation.
What is It
This section provides a brief discussion of
the lesson. This aims to help you discover
and understand new concepts and skills.
What’s More
This comprises activities for independent
practice to solidify your understanding and
skills of the topic. You may check the
answers to the exercises using the Answer
Key at the end of the module.
What I Have Learned
This includes questions or blank
sentence/paragraph to be filled in to process
what you learned from the lesson.
What I Can Do
This section provides an activity which will
help you transfer your new knowledge or
skill into real life situations or concerns.
Assessment
This is a task which aims to evaluate your
level of mastery in achieving the learning
competency.
Additional Activities
In this portion, another activity will be given
to you to enrich your knowledge or skill of
the lesson learned. This also tends retention
of learned concepts.
Answer Key
This contains answers to all activities in the
module.
At the end of this module you will also find:
The following are some reminders in using this module:
1. Use the module with care. Do not put unnecessary mark/s on any part of
the module. Use a separate sheet of paper in answering the exercises.
2. Don’t forget to answer What I Know before moving on to the other activities
included in the module.
3. Read the instruction carefully before doing each task.
4. Observe honesty and integrity in doing the tasks and checking your
answers.
5. Finish the task at hand before proceeding to the next.
6. Return this module to your teacher/facilitator once you are through with it.
If you encounter any difficulty in answering the tasks in this module, do not
hesitate to consult your teacher or facilitator. Always bear in mind that you are
not alone.
We hope that through this material, you will experience meaningful learning
and gain deep understanding of the relevant competencies. You can do it!
References This is a list of all sources used in
developing this module.
What I Need to Know
This module was designed and written with you in mind. It is here to help you
master the Quadratic Inequalities. The scope of this module permits it to be used in
many different learning situations. The language used recognizes the diverse
vocabulary level of students. The lessons are arranged to follow the standard
sequence of the course. But the order in which you read them can be changed to
correspond with the textbook you are now using.
The module is divided into three lessons, namely:
Lesson 1 – Introduction: Quadratic Inequalities
Lesson 2 – Solving Quadratic Inequalities in One Variable
Lesson 3 – Solving Quadratic Inequalities in Two Variables
After going through this module, you are expected to:
1. illustrates quadratic inequalities;
2. finds the solution set of quadratic inequalities in one variable;
3. graphs quadratic inequalities in two variables; and
4. finds the solution set of quadratic inequalities in two variables graphically.
What I Know
Choose the letter of the best answer, to find out how much you already know about this module. Write the chosen letter on a separate sheet of paper.
1. What is the degree of a quadratic inequality? A. B. C. D.
2. Which of the following is a solution of ?
A. B. C. D.
3. The following has more than one solution EXCEPT
A. B.
C. D. 4. Which of the following mathematical sentence is a quadratic inequality?
A. B.
C. D. 5. The following are quadratic inequalities EXCEPT
A. ( )( ) B. ( )
C. D. 6. Which of the following is the solution set of ?
A. * + B. * + C. * + D. * +
7. Which of the following points is a solution of ? A. ( ) B. ( ) C. ( ) D. ( )
8. The following points are solutions of EXCEPT A. ( ) B. ( ) C. ( ) D. ( )
9. Which of the following graphs represent the solution set of ?
A.
B.
C.
D.
10. Which of the following quadratic inequalities has a solution set of
?
A. B. C. D.
11. If ( )( ) is positive, which statement must be true? A. B. C. D.
12. Which statement is true about the inequality ? A. Points on the parabola are
solutions. B. The vertex is at point (2, 3).
C. Point (0, 0) is a solution. D. Point (2, 2) is a solution. 13. The figure on the right shows the graph of
. Which of the following is true about its solution set?
I. All points along the parabola belong to the solution set of the inequality.
II. All points on the shaded region belong to the solution set of the inequality.
III. All points along the parabola as shown by the broken line do not belong to the solution set of the inequality.
A. I and II B. II and III C. I and III D. I, II, and III
14. Which quadratic inequality is graph below?
A.
B.
C. ( )
D. ( )
15. Which graph shows the inequality ? A. B.
C. D.
𝑚
( 𝑟 )(𝑟 )
𝑥 𝑥
( 𝑛 )( 𝑛 )
𝑘 𝑘
𝑑 𝑠 𝑠
Lesson
1 Introduction: Quadratic Inequalities
Inequality tells us that the quantities associated are not balance. In studying
Mathematics, we know that it is not always “equal”, sometimes something is
greater or less than. As you go through this lesson, your knowledge of the different
mathematical concepts and skills in performing mathematical operations will help
you understand quadratic inequalities.
What’s In
Activity 1.1: What Satisfies Me?
Directions: Find the solution/s of the following mathematical sentences. Describe
these mathematical sentences.
1. 4.
2. 5.
3. 6.
What’s New
Activity 1.2: Who I Am?
Directions: Categorize the mathematical sentences below using the given table. Use
a separate sheet of paper.
𝑥 𝑥
𝑥 𝑥
( 𝑟 )(𝑟 )
𝑠 𝑠
𝑑
𝑥 𝑥
𝑘 𝑘
What is It
A quadratic inequality is an inequality sentence that contains a polynomial
of degree 2, and inequality symbols and are used.
Symbol Words Example
greater than
less than
greater than or equal to
less than or equal to
In Activity 1.2: Who I Am? a variety of mathematical sentences are given
where some of it are quadratic inequalities while the rest are quadratic equations or
none of the other two.
Quadratic Equations Quadratic Inequalities
Not a Quadratic
Equation nor a Quadratic
Inequality
These are quadratic
inequalities because the
degree is two and contain an
inequality symbol.
These are quadratic
equations because the
degree is two and has an
equal sign.
𝑚
( 𝑛 )( 𝑛 )
These are not a quadratic
inequality since the degree is
not 2.
What’s More
Activity 1.3: QI Builder…
Directions: Transform the following mathematical sentences into quadratic
inequalities by changing or adding only one element from the given.
1. 6. 2. 7. ( )( ) 3. 8. ( ) 4. 9. 5. ( )( ) 10.
Lesson
2 Solving Quadratic Inequalities in One Variable
What’s New
Activity 2.1: Carpentry: For Real?
Problem: A carpenter will build a house which has a floor area that is greater than
. He was instructed that the house to be built has a length which is 2 meters
longer than its width.
1. What is the mathematical sentence to be used in the given problem?
2. What is the possible dimension of the house to be built?
3. How did you come up with your answer?
What is It
A quadratic inequality in one variable can be written in one of the
following forms, where and are real numbers and .
let: width = 𝑤
length = 𝑤
𝐴𝑟𝑒𝑎 𝑙𝑒𝑛𝑔𝑡 ×𝑤𝑖𝑑𝑡
𝐴𝑟𝑒𝑎 𝑚
a. Therefore the mathematical sentence
is 𝑤 𝑤 .
b. The possible dimension of the house is 𝑚 × 𝑚,
𝑚 × 𝑚 and so on.
A quadratic inequality in one variable is very useful in solving real-life
problems such as posted above in Activity 2.1: Carpentry: For Real?
Solution:
The carpenter was instructed to build a house whose length is longer
than its width.
In solving quadratic inequalities in one variable, it is important that you
mastered your skills in solving quadratic equations which you’ve learned already in
the previous modules. To make this topic easier to understand and perform here
are the steps to follow in solving quadratic inequalities in one variable.
1. Change the quadratic inequality to quadratic equation in standard form.
2. Find the roots of its corresponding equality, using any of the following methods by extracting square roots, by factoring, by completing the square and by using quadratic formula.
3. Create intervals using the roots as critical x-values in dividing the number line into parts.
4. Test a number from each interval against the inequality. 5. Test the roots against the inequality. 6. Plot the roots on the number line. Solid circles are used in the graph
if they are part of the solution set and open circles if not a solution. Darken the parts of the number line that represents the solution set.
7. Write the solution set of the inequality.
Example: Find the solution set of the inequality .
Solution:
Step 1: Change to its
corresponding equality.
Step 2: Find the roots of
.
( )( ) By Factoring
Zero Product Property
Addition Property of Equality
Step 3: Create intervals.
(𝑤 )(𝑤 ) 𝑤 𝑤
(𝑤 )𝑤 𝑤 𝑤
𝑤 𝑤
𝒘 𝟕 𝒘 𝟓
Note: Length and width is always positive therefore, the width of the house is greater than 𝑚.
Step 4: Test a number from each
interval against the
inequality.
a. For , let
( ) ( )
TRUE
b. For , let
FALSE
c. For , let
TRUE
Step 5:Test also the points and against the inequality.
a. For
( ) ( ) FALSE b. For
FALSE
Step 6: Plot the roots on the number line. Darken the parts of the number line that represents the solution set.
Step 7: Write the solution set of the inequality.
The solution set of is
* +.
What’s More
Activity 2.2: Do You Know My Solution?
Directions: Find the solution set of the following quadratic inequalities then graph.
1.
2.
3.
4.
5.
Lesson
3 Solving Quadratic Inequalities in Two Variables
What’s New
Activity 3.1: Describe Me…
Directions: Using the graphs below answer the questions that follow.
1. Why is that the first graph has no shaded part?
2. In graph 2, 3, 4 and 5, what does the shaded region represent?
3. What is the difference between the broken line and solid line in sketching
the parabola?
Graph 1 Graph 2 Graph 3
Graph 5 Graph 4
What is It
Quadratic inequalities in two variables can be written in any of the
following forms, where and are real numbers and .
Examples:
1. 3.
2. 4.
The graphs in Activity 3.1: Describe Me… are all quadratic. The first graph
is of quadratic equation that is why it has no shaded part unlike the other graphs.
Graphs 2, 3, 4, and 5 are of quadratic inequalities, the shaded part represent the
solution set of the inequality. Broken line is use in outlining the parabola when
the inequality symbol is or , this indicates that the points on the parabola are
not part of the solution set. But, when the inequality symbol is or a solid line
is use in sketching the parabola, indicating that the points on the parabola satisfies
the given inequality.
In solving quadratic inequality in two variable it is necessary that you know
how to graph quadratic equations, which you learned already in the previous
modules. The solution set of a quadratic inequality in two variables can be
determined graphically. To do this, let us follow the steps below.
1. Write the inequality as an equation.
2. Graph the corresponding quadratic equation.
3. Choose a point ( ) in each region and check whether the given
inequality is satisfied.
4. Shade the entire region that satisfies the given inequality.
Example: Find the solution set of .
Solution:
Step 1: Write to its
corresponding equality.
Step 2: Graph .
Since the inequality symbol
is in the given inequality
a broken line will be used
in outlining the parabola.
Step 3: Select one point in each region and check whether the given inequality is satisfied.
a) Test point ( ). (Outside the parabola)
( ) FALSE
b) Test point ( ). (Inside the parabola)
( )
TRUE
Step 4: Shade the region inside the
parabola.
Therefore, all the points in the shaded region
are solution set of .
What’s More
Activity 3.2: Do I Satisfy You?
Directions: Determine whether the following points is a solution of the inequality
or not. Justify your answer.
1. ( ) 6. ( )
2. ( ) 7. ( )
3. ( ) 8. ( )
4. ( ) 9. (
)
5. ( ) 10. (
)
Activity 3.3: Do You Know My Solution?
Directions: Sketch the graph of the following quadratic inequalities and identify
three points of its solution set.
1.
Solution: _________________
2.
Solution: _________________
What I Have Learned
A. Describe the following and give at least three examples of each.
B. Describe and correct the error in graphing in the following figures.
1.
2.
Quadratic
Inequalities
Quadratic
Inequalities in
One Variable
Quadratic
Inequalities in
Two Variables
What I Can Do
Directions: Read the situation below then answer the questions that follow.
Popoy’s mother Aling Felomina told him to prepare a rectangular plot for a
vegetable garden. The length of the garden should be 15 feet longer than its width
and the area is less than 126 square feet.
1. How would you represent the width and length of the garden?
2. Give the mathematical sentence that represent the given situation.
3. What are the possible dimensions and areas of the garden?
4. Would it be realistic for the garden to have an area of 15 square feet? Justify
your answer.
Assessment
Choose the letter of your answer. Write the chosen letter on a separate sheet of
paper.
1. Which of the following is a quadratic inequality?
a. ( ) b. c. d.
2. The following are quadratic inequalities in two variables EXCEPT
a. ( ) ( ) b. ( ) c. d.
3. The following are quadratic inequalities EXCEPT
a. b. ( ) c. d. ( )
4. Which of the following is a solution of ? a. ( ) b. ( ) c. ( ) d. ( )
5. The following points satisfy the quadratic inequality EXCEPT a. ( ) b. ( ) c. ( ) d. ( )
6. Which of the following value of will satisfy ? a. b. c. d.
7. The following values of can satisfy EXCEPT a. b. c. d.
8. Which of the following is the graph of the solution set of ?
a.
b.
c.
d.
9. Which quadratic inequality is graphed below?
a. b. c. d.
10. What is the solution set of the inequality ?
a. {
} b. {
}
c. {
} d. {
}
11. Which quadratic inequality has a solution set * +?
a. b. c. d.
12. The following statement is true about the graph of the inequality EXCEPT
a. Point ( ) is a solution. b. The parabola opens upward. c. Points on the parabola are not a solution. d. The shaded region is inside the parabola.
13. If ( )( ) is negative, which statement is true? a. b. c. d.
14. Allen graph the quadratic inequality
as shown on the right. How are you going to correct her mistake?
a. A broken line shall be used in outlining the parabola.
b. The shaded part must be outside the parabola.
c. The vertex of the parabola must be
at ( ) and it opens upward. d. A broken line must be used in
outlining the parabola and the shaded part is outside it.
15. Which graph shows the inequality ?
a. b.
c. d.
Additional Activities
Directions: Compare and contrast the graphs of the following pairs of quadratic
inequalities.
1. 2. 3. 4.
Answer Key
References
Mathematics Grade 9, Learner’s Material, First Edition, 2014, pp. 96-113
Assessment
1.B 2.A 3.D 4.A 5.D 6.A 7.D 8.B 9.D 10.C 11.A 12.D 13.C 14.B 15.A
What's More
Activity 3.3
(Answer may vary.)
1.Sol. ( ) ( ) ( )
2.Sol. ( ) ( ) ( )
What's More
Activity 2.2
1.𝑥 𝑜𝑟 𝑥 2.𝑥 𝑜𝑟𝑥 3. 𝑥
4.
𝑥
5.𝑥 𝑜𝑟 𝑥
Activity 3.2
Solutions:𝐴 𝐵 𝐹 𝐺 𝐼 𝑎𝑛𝑑 𝐽
𝑚
What's More
Activity 1.3
(Answer may vary.)
1.𝑥 𝑥
2. 𝑥 𝑥
3. 𝑡 𝑡
4. 𝑥 𝑥 𝑥
5.(𝑎 )(𝑎 ) 6. 𝑥 𝑥
7.(𝑏 )( 𝑏 ) 8.𝑔(𝑔 ) 9.
What's In
1.𝑏 2.𝑐 3.𝑚 4.𝑟 𝑟 5.𝑚 ± 6.𝑛 𝑛
What I Know
1.B 2.A 3.C 4.C 5.B 6.A 7.B 8.D 9.A 10.A 11.C 12.D 13.B 14.A 15.A
DISCLAIMER
This Self-Learning Module (SLM) was developed by DepEd SOCCSKSARGEN
with the primary objective of preparing for and addressing the new normal.
Contents of this module were based on DepEd’s Most Essential Learning
Competencies (MELC). This is a supplementary material to be used by all
learners of Region XII in all public schools beginning SY 2020-2021. The
process of LR development was observed in the production of this module.
This is version 1.0. We highly encourage feedback, comments, and
recommendation.
For inquiries or feedback, please write or call: Department of Education – SOCCSKSARGEN Learning Resource Management System (LRMS)
Regional Center, Brgy. Carpenter Hill, City of Koronadal
Telefax No.: (083) 2288825/ (083) 2281893
Email Address: [email protected]
