8

Mathematics
Quarter 2 – Module 1:
“Differentiating Linear
Inequalities and Linear
Equations in Two
Variables”
CO_Q2_Mathematics8_M1
Mathematics – Grade 8
Alternative Delivery Mode
Quarter 2 – Module 1: Differentiating Linear Inequalities and Linear
Equations in Two Variables
First Edition, 2020

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Published by the Department of Education

Secretary: Leonor Magtolis Briones
Undersecretary: Diosdado M. San Antonio

Development Team of the Module

Writer: Anacita C. Suarez

Language Editor: Vicente P. Balbuena
Content Editors: Willy C. Dumpit, Vincent R. Lagat, Karen M. Basut
Layout Editor: Jay R. Tinambacan
Reviewers: Rhea J. Yparraguirre, Roche Y. Ocaña, Ruby V. Banias, Chielo D.
Palawan, Brian P. Atenta
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Layout Artists: Anacita C. Suarez, Ivan Paul V. Damalerio, Jake D. Fraga
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Office Address: Learning Resource Management Section (LRMS)

J.P. Rosales Avenue, Butuan City, Philippines 8600
Telefax No.: (085) 342-8207 / (085) 342-5969
E-mail Address: caraga@deped.gov.ph
 8
Mathematics
Quarter 2 – Module 1:
“Differentiating Linear
Inequalities and Linear
Equations in Two
Variables”

Introductory Message
This Self-Learning Module (SLM) is prepared so that you, our dear learners, can
continue your studies and learn while at home. Activities, questions, directions,
exercises, and discussions are carefully stated for you to understand each lesson.

Each SLM is composed of different parts. Each part shall guide you step-by-step
as you discover and understand the lesson prepared for you.

Pre-tests are provided to measure your prior knowledge on lessons in each SLM.
This will tell you if you need to proceed on completing this module or if you need
to ask your facilitator or your teacher’s assistance for better understanding of the
lesson. At the end of each module, you need to answer the post-test to self-check
your learning. Answer keys are provided for each activity and test. We trust that
you will be honest in using these.

In addition to the material in the main text, Notes to the Teacher are also
provided to our facilitators and parents for strategies and reminders on how they
can best help you on your home-based learning.

Please use this module with care. Do not put unnecessary marks on any part of
this SLM. Use a separate sheet of paper in answering the exercises and tests.
And read the instructions carefully before performing each task.

If you have any questions in using this SLM or any difficulty in answering the
tasks in this module, do not hesitate to consult your teacher or facilitator.

Thank you.

iiiiiiiiii
 What I Need to Know

This module is designed to help you differentiate linear inequalities and linear
equations in two variables. You are provided with varied activities to process the
knowledge and skills learned and to deepen and transfer your understanding of
the lesson. The scope of this module enables you to use it in many different
learning situations. The lesson is 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.

This module contains:

Lesson 1: Linear Equations and Inequalities and Linear

Equations in Two Variables

After going through this module, you are expected to:

1. distinguish the linear inequalities in two variables from linear equations in two
variables;
2. translate mathematical phrases into mathematical statements of linear
equations and inequalities in two variables; and
3. cite real life situations that can be represented by linear equations and
inequalities in two variables.

1
 CO_Q2_Mathematics8_M1

What I Know

Pre-Assessment

Directions: Choose the letter that corresponds to your answer. Write your
answer on a separate sheet of paper. After taking the test, take note of the item
that you were not able to answer correctly. Find the answer as you go through
this module.

1. The following expressions are examples of linear inequalities in two variables

A. – 𝑥 C. 13𝑥 − 4𝑦 = 21
EXCEPT:

B. D.13𝑥 − 24𝑦 < 32

A. 𝑥 < 𝑦 + 5 C. 𝑥
2. Which of the following mathematical expressions is written in standard form?

B. 2𝑥 + 3𝑦 < 12 D. x – 3y – 7

3. The mathematical statements below are all linear inequalities in two variables
EXCEPT:
3
A. < 10 − 2𝑥
C.

2𝑥 – 𝑦 > 6
D. 𝑦
B.

4. Which of the following verbal phrases DOES NOT represent a linear inequality in
two variables?
A. “the total number of male and female students in the learning center is
20”
B. “the total number of male and female students in the learning center is
at most 20”
C. “the total number of male and female students in the learning center is
greater than 20”
D. “the total number of male and female students in the learning center is
less than or equal to 20”

5. Which of the following is a linear inequality in two variables?

2
 A. 4𝑎 – 3𝑏 > 5 C. 𝑥 B. 7𝑐 + 4 < 12 − 3𝑑2 D.

A. 𝑥 C. 𝑥
6. Which inequality represents “the sum of x and y is at most 15”?

B. 𝑥 D. 𝑥

CO_Q2_Mathematics8_M1

7. Which of the following is true about the graphical solutions of a linear inequality
in two variables? A. It is a line.
B. It is a curve.
C. It is region of points bounded by a curve. D. It is a region
of points bounded by a line.

A. 𝑥 + 𝑦 > 29 C. 𝑥
8. The statement “the sum of two numbers is at least 29” can be expressed as:

B. 𝑥 + 𝑦 < 29 D.𝑥

9. By which condition does the symbol of a linear inequality in two variables

reverses its directions?
A. adding or subtracting negative real numbers
B. adding or subtracting positive real numbers
C. multiplying or dividing positive real numbers
D. multiplying or dividing negative real numbers

10.Which of the following is true about the graph of linear inequalities?

A. It is a line. C. It is a parabola. B. It is a curve. D. It is a half-plane.

11.Which of the following represents the graph of linear inequality in two

variables?

A. C.

3 CO_Q2_Mathematics8_M1
B. D.

4
12.Which statement best describes the graph of a linear equation in two
variables?
A. It is a line.
B. It is a curve.
C. It is region of points bounded by a curve. D. It is a region of points
bounded by a line.

13.James is asked what inequality symbol to be used to translate at most in

at most 15. His answer is <. Is he correct?

the mathematical sentence 5 times a number y plus 2 times a number x is

A. Yes, because the symbol for at most is .

B. Yes, because the symbol for at most is .
C. No, because the symbol for at most is .
D. No, because the symbol for at most is .

14.How does the graph of a linear inequality in two variables look like? A. It is
a line.
B. It is a curve.
C. It is region of points bounded by a curve. D. It is a
region of points bounded by a line.

15.John is multiplying the −2 to the linear inequality of two variables,

and his answer is . Is he correct?
A. Yes, because the product of the terms is correct, and the direction
of the inequality symbol is reversed.
B. Yes, because the product of the terms is correct, and the direction
of the inequality symbol does not change.
C. No, because multiplying a negative integer reverses only the
direction of the inequality symbol.
D. No, because multiplying a negative integer does not reverse the
direction of the inequality symbol.

5 CO_Q2_Mathematics8_M1
 Lesson Differentiating Linear
1 Inequalities and Linear
Equations in Two
Variables

Let’s start this module by recalling how to transform linear equations in two
variables in its standard form.

What’s In

Activity: Express Me

standard form 𝐴𝑥 + 𝐵𝑦 = 𝐶. Write your answer in a separate sheet.

Directions: Express the following linear equations in two variables into its

Given Standard Form

1. 2𝑥 = 𝑦 − 3

2. 𝑦 = −4𝑥 + 7

3. 2𝑦 = 8𝑥 − 9

4. 𝑥 = 2𝑦 + 4

5. 𝑥 – 3𝑦 – 7 = 0

Questions:

6 CO_Q2_Mathematics8_M1
 1. How did you find the activity? 2. How did you transform the
given equation into its standard form?

What’s New

Activity: Be my partner!

Directions: Match the verbal statement in column A to the mathematical

statement in Column B. Write your answers on a separate sheet of paper.

A B

______1. Fourteen more than a number x A. 14 – 𝑥 < 𝑦

is greater than 24
______2. Seven increased by a number x B. 2(𝑥 + 7) = 9
is equal to y
______3. Twice the sum of a number x and C. 14 + 𝑥 > 24
seven is 9
D. 7 + 𝑥 = 𝑦
______4. Seven more than the
product of fourteen and a

E. 14𝑥 + 7 ≤ 18
number x is less than or
______5. equal to 18

F. 7 + 𝑥 > 𝑦
A number x subtracted from
fourteen is less than y

Questions:

a. How did you find the activity?

b. What did you observe with the symbols used in each mathematical
statement?
c. What is the difference between symbol “=” from the symbol “≥”?
d. When shall you use the symbols ≥ and ≤? How about symbols “>” and
“<”?
e. When do you use symbol “=”?
f. What do you call mathematical statements a and d? How about b, c, e?

7 CO_Q2_Mathematics8_M1
 What is It

Equations and inequalities are two significant concepts in mathematics

that are related but are different in some ways. Inequality is a mathematical

, 𝑜𝑟 .
statement where one expression is not equal to another. It uses the symbols <,
>,
While equation uses the symbol “=” indicating that the value of the
expressions from both sides are equal.

The table below defines Linear Equations and Linear Inequalities in two
variables. See how these two differ from each other under several conditions.

Point of
Linear Equation Linear Inequality
Differences
A linear equation in two

standard form of 𝐴𝑥 +
variables is written in the
A linear inequality in two
𝐵𝑦 + 𝐶 = 0, where variables is formed when
𝐴, 𝐵, and 𝐶 are real
Definition symbols other than equal to,
such as greater than or less
𝑥
numbers and the
than are used to relate two

and 𝑦, represented by
coefficients of expressions, and two variables

𝐴 and 𝐵 respectively,
are involved.

are not equal to zero.

𝐴𝑥 𝐵𝑦 𝐶
𝐴𝑥 𝐵𝑦 𝐶
𝐴𝑥 + 𝐵𝑦 = 𝐶 𝐴𝑥 𝐵𝑦
𝐶 𝐴𝑥 𝐵𝑦 𝐶
Standard Form

𝐴𝑥 𝐵𝑦 𝐶
Symbols Used Symbol “Read as” Symbol “Read as”
“is less than”,
“is below”,
< “is smaller than”

“is greater than”,

> “is above”,
“is more than”

8 CO_Q2_Mathematics8_M1
 “is less than or
= “is equal equal to”,
to”, ≤ “is at most”,
“equals to”, “is not to exceed”,
“is”,
“is maximum”,
“equals”

≥ “is greater than

or equal to”, “is
at least”,
“is minimum”

≠ “is not equal to”

1. The sum of a number 𝑥
number 𝑥 and a and a number 𝑦 is greater
1. The sum of a

number y is 24. than 24.

𝑥 + 𝑦 = 24 𝑥 + 𝑦 > 24
2. A number 𝑥 decreased by
decreased by 𝑦 is a number 𝑦 is less than
2. A number x

24. 24.

𝑥 – 𝑦 = 24 𝑥 – 𝑦 < 24
Translating
Verbal
a number 𝑥 and number 𝑥 and thrice a
3. The sum of twice 3. The sum of twice a
Statements to
thrice a number 𝑦 number 𝑦 is at least 30.
Mathematical
is 30.
Statements
2𝑥 + 3𝑦 ≥ 30
2𝑥 + 3𝑦 = 30
4. Twice a number 𝑥
4. Twice a number 𝑥
number 𝑦 is at most 30.
decreased by thrice a

thrice a number 𝑦
decreased by

is 30. 2𝑥 – 3𝑦 ≤ 30

2𝑥 – 3𝑦 = 30
Characteristics of
Straight line Plane or half-plane
the graph

9 CO_Q2_Mathematics8_M1
 𝑦=−𝑥+1 𝑦>−𝑥+1

Sample graphs 𝑦 ≥ −𝑥 + 1

region of points bounded

Graphical solutions set of points on the line
by a line
Effects when
multiplied or Equality symbol is not Direction of the inequality
divided by a changed symbol is reversed
negative integer

Example: 1. Multiplying with a Negative Integer

Linear Equation: 3𝑥 – 2𝑦 = 6

3𝑥 – 2𝑦 = 6 Given.

3𝑥 + (−3𝑥) – 2𝑦 = (−3𝑥) + 6
with −3𝑥
Add both sides

0 – 2𝑦 = (−3𝑥) + 6

– 2𝑦 = −3𝑥 + 6
1 = 1

(− )(−2𝑦) (−3𝑥 + 6)(− )

Multiply both

2 2 sides by −

1 = (−3𝑥)(−) Distribute −

+ ( 6)(−

)2

𝑦10 = 3𝑥

−3
Simplify.
CO_Q2_Mathematics8_M1
 (−)(−2𝑦)
2

2
Equality
symbol does

Linear Inequality: 3𝑥 − 2𝑦 > 6

not change

3𝑥 – 2𝑦 > 6 Given.

3𝑥 + (−3𝑥) – 2𝑦 > (−3𝑥) + 6

with −3𝑥
Add both sides

0 – 2𝑦 > (−3𝑥) + 6

– 2𝑦 > −3𝑥 + 6

>

(− )(−2𝑦) (−3𝑥 + 6)(− ) sides by −

Multiply both

>

−
Distribute

(−)(−2𝑦) (−3𝑥)(−)

+ ( 6)(−)

Simplify.

𝑦 < 𝑥
−3 Direction
of the
inequality
symbol is
reversed

2. Dividing with a Negative Integer

Linear Inequality: 3𝑥 − 2𝑦 > 6

3𝑥 – 2𝑦 > 6 Given.

3𝑥 + (−3𝑥) – 2𝑦 > (−3𝑥) + 6

with −3𝑥
Add both sides

11 CO_Q2_Mathematics8_M1
 0 – 2𝑦 > (−3𝑥) + 6

– 2𝑦 > −3𝑥 + 6

𝑦 > 𝑥
( ) +
Divide both sides

by −2

Simplify.

𝑦< 𝑥 −3 Direction of the inequality

symbol is
reversed

What’s More

Activity 1: Sort me well!

Directions: Below are mathematical statements. Classify these statements in

the column where they belong. Write your answer in a separate sheet.

𝑦 = 7𝑥 + 21 𝑦 ≤ 7𝑥 + 21 3𝑦 − 7 = 5𝑥
3𝑦 – 7 < 5𝑥 10 – 5𝑦 = 7𝑥 10 – 5𝑦 ≥ 7𝑥
𝑦 = 5𝑥 + 20 3𝑥 + 4𝑦 < 15 3𝑥 + 4𝑦 = 15

𝑦 > 5𝑥 + 20

Linear Inequality in two variables Linear equations in two variables

Questions:

1. Which mathematical statements are linear inequalities in two variables?

Linear equations in two variables?

12 CO_Q2_Mathematics8_M1
 2. How did you identify linear inequalities in two variables and linear
equations in two variables?
3. In what way does 𝑦 ≤ 7𝑥 + 21 different from 𝑦 = 7𝑥 + 21? How about 3𝑦
– 7 < 5𝑥 and 3𝑦 – 7 = 5𝑥?
4. How do you differentiate linear inequalities in two variables from linear
equations in two variables?

Activity 2: Name Me!

Directions: Identify whether the situation represents a linear inequality in two

variables or not. Write LI if it is, otherwise write NLI. Write your answer in
separate sheet of paper.

1. The difference of the number of a 50 –peso tickets (𝑡) and 75 –peso tickets
(𝑠) is not equal to 200.
2. The price of a refrigerator (𝑟) is greater than the price of a washing
machine (𝑤) increased by Php850.
3. The number of girls (𝑔) in the theater arts club is 3 more than twice the
number of boys (𝑏).
4. A dozen of oranges (𝑜) added to two dozen of apples (𝑎) has a total cost of
at most Php 1, 950.
5. The number of red marbles (𝑟) is more than twice the number of yellow
marbles (𝑦).

Questions:

1. Which of the statements represent linear equations in two variables?

Which statements represent linear inequalities in two variables?
2. How did you identify if the statement represents a linear equation in two
variables? How about the statement of linear inequalities in two variables?

Activity 3: What am I?

Directions: Translate mathematical statements into mathematical sentences of

linear equations and linear inequalities in two variables. Write your answer on a
separate sheet.

Mathematical Mathematical Sentences

Statements
1. 𝑦

2. 𝑦

13 CO_Q2_Mathematics8_M1
 3. 𝑦

4. 𝑦
5.

Questions:

1. Which of the statements represent linear equations in two variables? Which

statements represent linear inequalities? 2. How did you translate
mathematical statements into mathematical sentences?

Activity 4. Shall I Stay or Be the Other Way?

Directions: Write the resulting mathematical statements after applying the

condition specified in each item. Write your answers on a separate sheet.

Resulting
Mathematical
Condition Mathematical
Statements
Statements
1. Multiply both sides by

2. Multiply both sides by

Multiply both by

3.
4. Divide both sides by
5.
Divide both sides by

What I Have Learned

Remember Me!

14 CO_Q2_Mathematics8_M1
Directions: Fill in the blank with an appropriate word or phrase. Write your
answer in a separate sheet of paper.

A mathematical statement where one expression is not equal to another

(greater than), < (2_________), (3____________), (is less than or equal to), and
expression is called an 1__________. The usual symbols of inequality are >

(is not equal to). On the other hand, = (equal) is the symbol used for linear
equations in two variables.
A Linear 4______________ in two variables can be written in one of the

𝐴𝑥 + 𝐵𝑦 𝐶
following forms:

𝐴𝑥 + 𝐵𝑦 𝐶
𝐴𝑥 + 𝐵𝑦 𝐶
𝐴𝑥 + 𝐵𝑦 𝐶

Both linear equations and linear inequality in two variables can also be
presented through graph. The graph of linear inequality is a 5____________ or a
6__________. On the other hand, the graph of a linear equation is a 7____________.

When a linear 8____________ in two variables is multiplied or divided by a

negative integer, the equality symbol does not change. However, when a linear
inequality in two variables is 9__________ or 10__________ by a negative integer,
the direction of the inequality symbol changes.

What I Can Do

Words Are All I Have

Directions: Write a poem/spoken poetry describing the differences of linear

equations and inequalities in two variables.

Rubric: Poem/Spoken Poetry Piece

10 8 6 4

The literary piece The literary piece The literary piece The literary piece
contains at least 5 contains at least 3 contains at least 2 contains only 1
differences of differences of differences of difference of linear
linear equations linear equations linear equations equations and

15 CO_Q2_Mathematics8_M1
 and linear and linear and linear
linear inequalities
inequalities in two inequalities in two inequalities in two
in two variables.
variables. variables. variables

Assessment

Post Assessment

Directions: Read the following questions carefully and choose the letter that
corresponds to your answer. Write your answers on a separate sheet of paper.

1. Which of the following is NOT a symbol of linear inequality?

A. C.
B. D.

2. What is the graph of linear inequality?

A. a half plane C. parabola
B. half of parabola D. straight line

3. Which is true about the graphical solution of inequalities in two variables?

A. region of points C. sets of planes
B. sets of points D. region of plane

A. 6𝑎 – 3𝑎 = 9 C. 𝑝 B. 𝑘 + 4 < 8 + 2𝑘
4. Which of the following shows linear inequality in two variables?
D.

5. Which of the following inequalities is the same as “the sum of 2𝑥 and 𝑦 is

at least ”?
A. C. B. D.

6. Which of the following is true about the graphical solutions of a linear

equation in two variables?
A. It has no solution.
B. It has only two solutions.
C. It is a set of points on a line.
D. It is a region of points bounded by a line.

16 CO_Q2_Mathematics8_M1
7. The mathematical statements below are all linear inequalities in two

A. 𝑥 – 𝑦
variables EXCEPT:

B. 𝑥 𝑛
C.
D.

8. Which of the following is true about the graph of linear equation in two
variables?
A. It is a plane. C. It is a parabola.
B. It is half of parabola. D. It is a straight line.

9. The following linear inequalities in two variables are in standard form

A. 𝑦 > 2𝑥 + 25 C.
EXCEPT

B. 𝑥 – 2𝑦 < 25 D.

10.The following represent the graph of linear inequality in two variables

EXCEPT

A. C.

B. D.

11.What condition/s reverses the direction of the inequality symbol in a linear

inequality?
A. adding negative integers to both sides of the expression
B. subtracting negative integers to both sides of the expression

17 CO_Q2_Mathematics8_M1
 C. multiplying or dividing positive integers to both sides of the
expression
D. multiplying or dividing negative integers to both sides of the
expression

12.How does the graph of a linear inequality in two variables look like? A. It is
a line.
B. It is a curve.
C. It is a region of points bounded by a line.
D. It is a region of points bounded by a curve.

13.Peter is multiplying to the linear inequality in two variables,

his answer is . Is he right?
A. Yes, because the product of the terms is correct and the direction of
the inequality symbol is reversed.
B. Yes, because the product of the terms is correct and the direction of
the inequality symbol does not change.
C. No, because multiplying a negative integer reverses only the
direction of the inequality symbol.
D. No, because multiplying a negative integer does not change the
direction of the inequality symbol.

14.Which of the following statements best describe a linear inequality in two

variables?
I. The graph is a set of points or a line
II. It involves the following symbols
III. The direction of the inequality symbol reverses when multiplied by
a negative integer

A. I & II C. II & III

B. I & III D. I, II & III

15.Kim is asked what inequality symbol to be used to translate at least in the

mathematical sentence 8 times a number y minus 3 times a number x is

30. Her answer is >. Is she correct?

at least

A. Yes, because the symbol for at least is .

B. Yes, because the symbol for at least is .
C. No, because the symbol for at least is . D. No, because the symbol
for at least is .

18 CO_Q2_Mathematics8_M1
 Additional Activities

Direction: Cite real-life situations in your community that represent linear

equations and inequalities in two variables.

Rubric:
10 8 6 4
At least 6 real- life 4-5 real- life 2-3 real- life Only 1 real- life
situations are situations are situations are situation is cited.
cited. cited. cited.

19 CO_Q2_Mathematics8_M1
 Answer Key References

Abuzo, Emmanuel P., et.al, Mathematics- Grade 8 (Learner’s Module First

Edition, 2013. Published by the Department of Education)216-219.

20 CO_Q2_Mathematics8_M1
Orgines, Fernando B.,Zenaida B. Diaz, Maharlika P. Mojica, Calina B.
Manalo, Josephine L. Suzara, Jesus P. Mercado, Mirla S. Esparrango,
Nestor V. Reyes,Jr., Next Century Mathematics 8 (Quezon
Ave.,Quezon City, PHOENIX Publishing House,2013) 370-371.

21 CO_Q2_Mathematics8_M1
For inquiries or feedback, please write or call:

Department of Education – Bureau of Learning Resource

Ground Floor, Bonifacio Building, DepEd Complex
Meralco Avenue, Pasig City, Philippines 1600

Telefax. Nos.: (632) 8634-1072; 8634-1054; 8631-4985

Email Address: blr.lrqad@deped.gov.ph * blr.lrpd@deped.gov.ph