Mathematics

Quarter 2 – Module
8
Week 1 – Week 4

Department of Education • Republic of the Philippines

Mathematics – Grade 8
Alternative Delivery Mode
Quarter 2 – Module: Week 1- 4
First Edition, 2020

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Published by the Department of Education

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Development Team of the Module

Authors: Flora S. Isada , Lanie S. Salamanes, Caren Lynn N. Diaz, Roderick Partos,
Beatrice Tarcena, Lorelie R.Huidem, Miriam F.Genoza

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 8

Mathematics
Quarter 2 – Week 1-4:

This instructional material was collaboratively developed and

reviewed by educators from public and private schools, colleges, and
or/universities. We encourage teachers and other education stakeholders to
email their feedback, comments, and recommendations to the Department of
Education at action@deped.gov.ph.

We value your feedback and recommendations.

Department of Education • Republic of the Philippines

Introductory Message

For the facilitator:

This module is designed to provide the facilitator with background knowledge and
understanding with these four basic components : (1) an appreciation of the discipline of
mathematics itself – what it means to “do mathematics” , (2) an understanding on how
students learn and construct ideas, (3) an ability to design and select tasks so that students
learn mathematics in a problem solving environment, and (4) the ability to integrate
assessment with the teaching process in order to enhance learning and improve daily
instruction. This gives an instruction for you to orient the learners and support the parents,
elder sibling etc. of the learners on how to use the module. Furthermore, this also instructs
you to remind the learners to use separate sheets in answering the pre-test, self-check
exercises, and post-test.).

For the learner:

To get the most out of this module, here are a few reminders:

1. Take your time in reading the lessons

2. Write down points for clarification. You may discuss these points with your teacher or
mentor.
3. Perform all activities and answer all exercises. The activities and/or exercises here are
designed to enhance your understanding of the ideas and concepts being discussed.
4. Answer all tests in this module, including the assessment and check your answers in
the answer key. These tests will give you idea, how well you understand the lessons.
5. As a courtesy to the next user, PLEASE DO NOT WRITE ANYTHING on any part of this
module. Write all your answers on a separate sheet of paper. These shall be part of your
formative evaluation.

This module is self- instructional and allows you to learn in your own space and pace. So, relax
and enjoy!

CONTENT STANDARDS: The learner demonstrates key concepts of linear inequalities in two
variables,systems of linear inequalities in two variables and linear functions.

PERFORMANCE STANDARDS: The learner is able to formulate and solve accurately real-life
problems involving linear inequalities in two variables, systems of linear inequalities in two
variables and Linear Functions.

The module is divided into nine lessons, namely:

 Lesson 1 – Linear Inequality in Two Variables

 Lesson 2 – Graphing Linear Inequalities in Two Variables.
 Lesson 3 - Solving Problems Involving Linear Inequality in Two Variables
 Lesson 4 - Solving System of Linear Inequality in Two Variables
 Lesson 5 - Solving Problems Involving System of Linear Inequalities
 Lesson 6- Relation and Function
 Lesson 7 – Determining the Dependent and Independent Variables
 Lesson 8 – Domain and Range of a Function

Department of Education • Republic of the Philippines

 Lesson 9 – Linear Function

After going through this module, you are expected to:

 Illustrates Linear Inequality in Two Variables

 Differentiates Linear Inequality in two variables from linear Equation in two variables
 Graphs linear inequalities in two variables.
 Solves problems involving linear inequality in two variables.
 Solves system of Linear Inequality in two variables
 Solves problems involving system of linear inequalities
 Illustrates a relation and a function
 Verifies if a relation is a function
 Determines the dependent and independent variables
 Writes the domain and range of a graph, rule, table of values and mapping
 Illustrates a linear function

Department of Education • Republic of the Philippines

Week 1 - Day 1-2

DAY 1

What I Know
Multiple Choice: Choose the letter of the best answer. Write the chosen letter on a
separate sheet of paper.
1. What is the slope in a linear inequality 2x – 3y < 6 ?
2 2 3 3
A. − 3 B. 3 C. − 2 D. 2
2. To solve an inequality means to find all values of the variable for which the statement is true.
A. Always True B. Sometimes True C. False D. Cannot be determined
3. What inequality represents the verbal expression? 8 less than number n is less than 11.
A. 11 – 8 < n B. n – 8 < 11 C. 8 – n < 11 D. 11 < 8 – n
4. What is the slope-intercept form of the given inequality, 8x+4y>32?
A. y > -2x+8y B y < -2x + 8 C. y > 2x-8 D. y < 2x – 8
5. Which of the following is a solution to y ≥ 2x + 5?
A. (1, 0) B. (2, -2 C. (-2, 2) D. (-1, -3)

WHAT’S IN
A Linear Inequality is like a Linear Equation (such as y = 2x+1) ...but it will have
an Inequality like <, >, ≤, or ≥ instead of an =.
A linear inequality in two variables
can be written in the following form.
�� + �� > � greater than
�� + �� < � less than
�� + �� ≥ � greater than or equal to,
at least
�� + �� ≤ � less than or equal to, at
most
�� + �� ≠ � not equal

Where a, b and c are real numbers. A solution of a linear inequality in two variables is an
ordered pair (x,y) that makes the inequality true.
We can also identify whether the given expression is a linear inequality or not.
Example 1 :
Identify if the given inequality is a linear inequality in two variables or not.
a. 2x + 3y > 5
b. x2 – y2 ≤ 1
c. x ≥ 6y + 3
d. y < 2xy - 3
Solution:
a. 2x + 3y > 5 Linear inequality
b. x2 – y2 ≤ 1 Not Linear inequality
c. x ≥ 6y + 3 Linear inequality
d. y < 2xy – 3 Not Linear inequality

Solution of a Linear Inequality

An ordered pair (x, y) is a solution of a linear inequality in x and y if a true statement results
when the values of x and y are substituted into the inequality.
Example 2. Tell whether the ordered pair is a solution of the inequality.
a. 2 x  y  3 ; (-1,9)
b. x  3 y  8 ; (2,-2)
Solution:
a. 2 x  y  3 Write the inequality
2(1)  9  3 substitute -1 for x and 9 for y
-2 + 9 < -3 simplify
7 < -3 NO 7 is not less than -3
 so (-1,9) is NOT a solution of the inequality
b. x  3 y  8 Write the inequality
2  3(2)  8 substitute 2 for x and -2 for y
2  6  8 simplify
8  8 TRUE, 8 is equal to 8  so (2,-2) is a solution to an inequality
Page 6 of 32
Activity 1: Identify if the given inequality is linear inequality in two variables. Draw a happy
face if it is linear inequality in two variables if not draw a sad face .

1. 2x2 + 2y < 1 6. 4y + 24x < 36

2. 4 > 4x + y2 7. y ≥ 6y + x
3. x – 3y ≤ 9 8. xy < 49
4. 2xy – 10x ≥ 25 9. 2x – 20y ≤ 10x
5. y2 – 6y > 9 10. 3 (x + y) ≥ 2x + 6

Activity 2:
Which of these ordered pairs is the solution of the following inequality.
1. 2x + y < 4
a. (2, 3) b. (-2, -4)
2. x + 3y > 9
a. (2, 3) b. (1, -4)
3. Determine if the given points are solutions of the inequality 3x - 2y > 12.
a. ( − 2, 4) b. (2, − 3) c. (5, 0)

DAY 2

WHAT’S NEW

Equations and inequalities are two important tools in Mathematics. The former expresses
identity or exactness of quantities, while the latter implies difference in quantities.

Point of Comparison
or Difference Linear Equation Linear Inequality

Symbol used = >, <, ≥ , ≤, ≠

Sample expression x + 2y = 3 x + 2y > 3
3x + y < 7
Geometric Intersecting coinciding or Plane or half plane
representation parallel lines,

Sample Graphs

WHAT IS IT

Activity1:
Classify the following
Parallel Lines Half-plane 5x + 3y = 2

6y ≠ 2x - 3 Set of Points Region of points

= < >
Linear Equality Linear Inequality

Activity 2:
Tell whether the following is LE (linear equation in two variables) or LI (linear inequality in
two variables):
1. x + 2y < 1 6. 4y + 24x = 36
2. 4 + 4x = y 7. 10 < 6y + x
3. y – 3x = 9 8. 7x – 14y ≠ 49
4. 4x > 10y + 25 9. 6y – 20x = 100
5. 3x – 4y ≤ -10 10. x = y + 10

Page 7 of 32
WHAT I HAVE LEARNED

In mathematics a linear inequality is an inequality which involves a linear function.

A linear inequality contains one of the symbols of inequality:[1]. It shows the data
which is not equal in graph form.
< less than > greater than
≤ less than or equal to ≥ greater than or equal to
≠ not equal to = equal to
A linear inequality resembles in form an equation, but with the equal sign replaced
by an inequality symbol. The solution of a linear inequality is generally a range of
values, rather than one specific value

ASSESSMENT

A. Choose the letter of the correct answer:

____1. Which inequality describes all points to the right of y-axis?
A. � ≤ 0 B. � ≥ 0 C. � ≥ 0 D. � ≤ 0
____2. How many solutions does a linear inequality in two variables have?
A. 0 B. 1 C. 2 D. many
____3. Determine if the given points are solutions of the inequality 3x - 2y ≥ 12.
A. ( − 2, 4) C. (2, − 2) B. (4, 0) D. (0,0)
____4. Which of the following is a solution to y ≥ 2x + 5?
A. ( 1, 0) C. (-2, 2) B. (2, -2) D. (-1, -3)
____5. Translate to an inequality: Three more than four times a number is less than 23.
A. 3x + 4 < 23 C. 4x + 3 < 23 B. 3 > 4x < 23 D. 3 > 4x -23
_____6. What inequality represents the verbal expression? 8 less than number
n is less than 11.
A. 11 – 8 < n B. n – 8 < 11 C. 8 – n < 11 D. 11 < 8 – n
____ 7. What is the slope-intercept form of the following inequality, 8x+4y>32?
A. y > -2x+8y B y < -2x + 8 C. y > 2x-8 D. y < 2x – 8
_____8. What is the slope in a linear inequality 2x – 3y < 6 ?
2 2 3 3
A. − 3 B. 3 C. − 2 D. 2

B. Identify if the following is a linear equation in two variables or a linear inequality in two
variables
1. 4x + 3y = 15 6. 4x + 2y < 10
2. 7x – y > 9 7. 10x – y = 9
3. y = 5x + 7 8. x = 5y + 7
4. 5y < 2x – 3 9. 11y < 2x - 3
5.10 – y ≠ x + 2 10. 12 ≠ x + y

Week 1 - Day 3

What I Know
Directions: Write a check mark (√) if the given coordinate is a solution to the
inequality and cross (x) if not.

1. y ≥ -3x + 4; (0, 2)
2. y < 2x – 1 ; (-1, 4)
3. y > -x – 5; (-5, 5)

B. Graph the given linear inequalities in two variables.

1. y ≥ -3x + 4 2. y ≤ 2x – 1

Page 8 of 32
 WHAT’S IN

In the preceding lessons, you learned that a linear inequality in two variables is an equality
that can be written in the following form:
Ax + By < C Ax + By ≤ C Ax + By > C Ax + By ≥ C

WHAT’S NEW

Activity: Let’s Explore!

Directions: You will need a highlighter or color for this activity.

1. Highlight all the points that make the inequality x + y ≤ 2 true.

2. Graph the line x + y = 2. What have you noticed about this line?
3. What does the less than or equal to (≤) symbol do to the graph?
4. The point (-3.5, -1.5) is also a solution to this inequality. Name 3 other points that are
solutions and mark them on the coordinate plane?
5. How many other points are in the solution set of this inequality? How can we show
these points on the coordinate plane?

WHAT’S IS IT

Steps in graphing linear inequalities in two variables:

1. Replace the inequality symbol with an equal sign. The resulting equation will become the
plane divider.
Example:
a. y > x + 4 y=x+4
2. Graph the resulting equation with a solid line if the original inequality contains ≤ or ≥
symbol. The solid line indicates that all points on the line are part of the solution of the
inequality. If the inequality contains < or > symbols, use a dashed or broken line. The dashed
or broken line indicates that all points on the line are not part of the solution of the inequality.
Example:
a. y > x + 4

3. Choose a point in one of the half-plane that is not on the line. Substitute the value of the
coordinate to the inequality. If the coordinate satisfy the inequality (True), shade the half-plane
where your test point is located. If the coordinate didn’t satisfy the equation (False), shade the
half-plane opposite your test point.

Location of Point Test Point Half-Plane to Shade

Above the line True Above the line
Above the line False Below the line

Location of Point Test Point Half-Plane to Shade

Below the line True Below the line
Below the line False Above the line

a. Test point: (1, 3)

y>x+4
3>1+4
3 > 5 False

Page 9 of 32
 Shaded
above
the line

WHAT’S MORE

Activity: Let’s Do This Step-by-Step

Directions: Graph the given linear inequalities in two variables using the step-by-step
procedure.
1. y ≤ -3x + 3
a. Change the inequality symbol to an equality symbol
b. Graph the inequality using the x and y intercepts
If x = 0 If y = 0
c. Kind of line to use (solid or dashed)
d. Graph

e. Is the test point (-1, 2) a solution (True) or not a solution (False)?

f. Shade the correct half-plane. Use the coordinate plane in d.

WHAT I HAVE LEARNED

Activity: Our Steps!

Directions: Make a conceptual map in graphing linear inequalities in two variables. Write
important concepts every step inside the box. If necessary, add another chevron to complete
your conceptual map.

Step 1

Step 2

Step 3

Step 4

Page 10 of 32
 WHAT I CAN DO

Due to covid-19
pandemic, Marian sells face
shield and face mask online.
She sells the face shield for
P35 each and face mask for
P20 each. She hopes to earn
at least P1200 this Saturday
so she can pay her monthly
bills.

1. Identify three combinations of face shield and face mask that will earn Marian more
than P 1200.
2. Identify three combinations of face shield and face mask that will earn Maria exactly P
1200.
3. Identify three combinations that will not earn Marian at least P1200
4. Graph your answers to problems 1 and 2 in the coordinate plane and then shade a half-
plane that contains all possible solutions to this problem.
5.

6. Create a linear inequality that represents the solution to this problem. Let x be the
number of face shield and let y be the number of face mask that Marian sells.

ASSESSMENT

Directions: Choose the letter of the best answer. Write the chosen letter on a separate sheet of
paper.

For numbers 1 – 2 write A if the given point is a solution of the inequality y < -x + 8.
1. (4, 0)
2. (10, 2)
3. If the linear inequality contains an equality symbol of < or >, what kind of line are
you going to use?
A. broken line B. solid line C. plane line D. coordinate line
4. If the linear inequality contains an equality symbol of ≤ or ≥, what kind of line are
you going to use?
A. broken line B. solid line C. plane line D. coordinate line
5.Which of the following shows the graph of the linear inequality y ≥ 4x - 1
A. B. C. D.

Page 11 of 32
Week 1 - Day 4

What I Know

TRANSLATE ME!
Directions: Write each statement as a linear inequality in two variables.
Situation Linear Inequality
1. The difference between the weight of Aira (a) and Nica (n) is at least 9.
2. Thrice the number of green marbles (g) is less than the number of
yellow marbles (y).
3. The number of bananas (b) more than twice the number of mangoes
(m) is greater than 28.
4. The total amount the family spends for electric bill (e) and water bill
(w) is at most Php 3,000.
5. Twice a number (x) increased by 9 is greater than another number
(y).

WHAT’S IT

Find out how linear inequalities in two variables are used in real-life situations and in solving
problems.
Certain situations in real-life can be modeled by linear inequalities in two variables.
Consider the following example.
1. The sum of 5-peso coins and 10-peso coins is greater than Php 180.
a. What mathematical statement represents the total amount of 5-peso coins and 10-
peso coins? Define the variables used.
Answer:
Let x = the number of 5-peso coins
y = the number of 10-peso coins
Linear Inequality: 5x + 10y > 180
b. Draw the graph of the inequality.
Answer: Refer to the graph at the right.
c. How many 5-peso coins and 10-peso coins are there? Give at least 3 ordered pairs
for the possible answers.
Answer:
Refer to the graph for the possible answers. You can only get solutions from the
shaded region. Locate some points.
1. First ordered pair: ( 5, 17 )
*There are 5 5-peso coins and 17 10-peso coins.

2. Second ordered pair: ( 8, 16 )

*There are 8 5-peso coins and 16 10-peso coins.

3. Third ordered pair: ( 12, 14 )

*There are 12 5-peso coins and 14 10-peso coins.

Page 12 of 32
 d. Suppose there are 11 10-peso coins, what is the minimum number of 5-peso coins?
Answer:
Given: y = 11
Substitute this value to the inequality.
5x + 10y > 180
5x + 10(11) > 180
5x + 110 > 180
Applying Subtraction Property of Inequality,
5x + 110 – 110 > 180 – 110
5x > 70
Applying Division Property of Inequality,
5x > 70
5 5
x > 14 .The minimum number of 5-peso coins is 15.
WHAT IS IT

Directions: Answer the following questions. Give your complete solutions or explanation.
●The difference between the height of Jasmine and Kimberly is less than 8 inches.
1. What mathematical statement represents the difference in the heights of
Jasmine and Kimberly? Define the variables used.
2. Based on the mathematical statement you have given, who is taller? Why?
3. Suppose Jasmine’s height is 5 ft and 1 in, what could be the height of Kimberly?
Explain your answer.

WHAT I HAVE LEARNED

Directions: Put a smiley mark in each column if you had learned the concept and this mark
if not.

1. I learned how to write a mathematical statement that

represents a real-life situation.
2. I learned how to define the variables I used to represent
the unknowns.
3. I learned how to solve linear inequality.
4. I learned how to answer specifically what is asked in the
problem.

Week 2 - Day 1 & 2

What I Know
Directions: Choose the letter of the best answer. Write the chosen letter on a
separate sheet of paper. Graph for no. 1

1. Using the graph at the right name the point that does NOT belong to the solutions
�−�≤�
set of ?
�+� > �
a. A = (3, 3) b. B = (6, 0) c. D = (5, 1) d. F = (1, 1)
�−�≤ �
2. Which of these is belongs to the solutions set to the system ?
�+�>�
a. 0, − 6 b. −4, 0 c. −3, − 8 d. 0, 6
3. How do you describe the boundary line representing the graph of 5 ≤ � ?
a. a broken line b. a combination of solid a c. a solid line d.
spiral
2� − �� < �
4. Which of these is one of the solutions to the system ?
�� > �� − �
a. 0, 0 b. 4, 0 c. −2, − 6 d. 5, 0
�+�<�
5. Which of these is one of the solutions to the system ?
��� + �� > ��
a. 0, 0 b. 4, 0 c. 3, 8 d. −8, 0
Page 13 of 32
 WHAT’S IN
To prepare you for the new concepts that will be discussed in this module, it is
important that you have mastered how to graph linear inequalities, wherein you have to
remember to graph the "equals" line, then shade in the correct area. or remember these three
steps:
1. Write the equation in y form so "y" is on the left and everything else on the right.
2. Draw the boundary line, "y=" line (make it a solid line for y≤ or y≥, and a dashed line
for y< or y>)
3. Shade above the line for a "greater than" (y> or y≥)or below the line for a "less than"
(y< or y≤).

WHAT’S NEW

Write your answer for the following activity on a separate sheet of paper.
Description: The aim of the activity is to show the common shaded region of two half-plane.
Activity 1: OVERLAPPING

Direction: Study the pattern to complete the last two squares of group D.

1 2 3 4 1 2 3 4

Questions:
1. How do you describe all the plane 3 (starting from left going to the right in each
group)? ____________________
2. Does it represent the shaded part of plane 1 and 2? (Yes/No) _____
3. How do you describe the plane 4?
4. What does the shaded part in this column represent? ____________________

WHAT’S IS IT

Solving a system of linear inequalities is very similar to solving systems of

equations. Graphing is the easiest way to solve linear inequalities.
What you did in activity 1 is locating the common shaded part of two inequalities,
since we describe the graph of inequalities as shaded half-plane. All plane 4 in activity 1
represent the exact common shaded part of plane 1 and plane 2. This common shaded part
is the regions of solution. To solve system of linear inequalities we need to find the region of
solutions.
WHAT I HAVE LEARNED

Description: The aim of the activity is to show the common shaded region of two half-plane.
Activity 3:
Direction: Fill in the blank with the correct letter to complete the words that satisfy each
statement. (write only one letter on top of each blank.)
1. A solution of a system of linear inequalities is any (S ___ ___ )of ordered pairs that satisfies
each linear inequality in the system.
2. To find the solution of the systems of linear inequalities, find the region where the graph
(__ __ __ __ __ __ P P __ __ )
3. To verify the solution of the systems of linear inequality pick and test a (P __ __ __ __ ) in
the region.
To find the solutions to the s system of linear inequalities in two variables graphically, we
simply
1. Draw graph of the (B __ __ __ __ __ __ __ ) line for the first inequality.
2. (S __ __ __ __) the region that satisfies the inequality,
3. Draw the ( __ __ __ __ __ ) of the boundary line for the second inequality,
4. Shade the region that (__ __ __ __ __ __ __ __ ___) the second inequality,
5. The solution set will be the ( O __ __ __ __ __ __ __ __ __ __ ) region.

WHAT I CAN DO

Description: This activity will enable you to visualize all the possible answer to Mr. Wash
Opening problem

Page 14 of 32
 Directions: Give at least 2 possible answers to the opening problem of Mr. Wash, graph all
inequalities in the given problem.
In the opening activity you have represent the two important condition that must be satisfy in
� + 2� ≤ 20
the given problem as a system .
� + � ≥ 15
Since Mr. Wash wants to use an alcohol at least 15 times a day, therefore this indicate
that
1. the number of times he uses AAV-a can be greater than or equal to zero
2. or the number of times he uses the AAV-b can be greater than or equal to
zero.
Represent this as
Inequality 3 � > _________
Inequality 4 � > _________
Graph the 4 inequalities that we need to consider in getting all the possible number of
times Mr. Wash use each AAV Machine. (use light shade only to graph each inequality)

� + 2� ≤ 20 Possible answers base on your

� + � ≥ 15 graph
� > ________
� > _________

ADDITIONAL ACTIVITY

Write your answer for the following problems in a graphing paper

I. Solve system of inequalities graphically. (use straight edge to draw the boundary line and
coloring material to shade region.)
2� � ≤1
� ≤− 3 2� + � ≤ − 5 � ≤ −5 � ≤ � − 4
1. � ≥− 3 2. � − 3� ≥ − 1 3. 3 4.
�+� ≥ −2 � ≤− 2
� + � <− 4 � < −4 2� − 3� ≥5
�> −4
II. Write a system of linear inequalities to describe the situation in each problem. Then
solve the system graphically.
1. The size of a basketball floor varies due to building sizes and other concerns such as
cost, etc. the length L is to be at most 94 ft., and the width w is to be at most 50 ft.
Graph a system of inequalities that describes the possible dimensions of basketball
floor.
2. Write four systems of four inequalities that describe a 3-unit-by-3-unit square that has
(0, 0) as one of the vertices.
3. Many elevators have a capacity of 1000 kg. Suppose that children (each weighing 35 kg.)
and adults (each 80 kg.) are inside an elevator. Graph a system of inequalities that
indicates when the elevator is overloaded.

Week 2 - Day 3 & 4

What I Know

Directions: Choose the letter of the best answer. Write the chosen letter on a separate
sheet of paper.
Ayie makes home decorations. It takes her 2 hrs to make a bouquet and 1 hr to make
a basket. She can work no more than 40 hrs per week. The cost to make one bouquet
Page 15 of 32
is 150 pesos and the cost to make a basket is 100 pesos. She can afford to spend no
more than 3,600 pesos per week.
1. Which of the following inequality will represent the number of hours that Ayie can
work?
a. 2� + � < 40 b. 2� + � ≤ 40 c. 2� + � > 40 d. 2� + � ≥ 40
2. Which of the following inequality will represent the amount that Ayie can spend?
a. 150� + 100� < 3600 c. 150� + 100� > 3600
b. 150� + 100� ≤ 3600 d. 150� + 100� ≥ 3600
3. Which of the following graphs is the graph of the system of inequality?

a. b

c. d.

4. What could be the reasonable number of baskets that Ayie can make?
a. 0 ≤ � ≤ 36 b. 0 ≥ � ≤ 36 c. � ≤ 36 d. � ≥ 36
5. What could be the reasonable number of bouquets that Ayie can make?
a. 0 ≤ � ≤ 20 b. 0 ≥ � ≤ 20 c. � ≤ 20 d. � ≥ 20

WHAT’S IN

In this module, the students will encounter the different problems that can be
solved using the system of linear inequalities in two variables. Their prior knowledge
about the basic topics they’d been discussed from their previous levels will also help
them to understand and answer the problem.

WHAT’S NEW
A system of linear inequalities in two variables consists of at least two
linear inequalities in the same variables. The solution of a linear inequality is the
ordered pair that is a solution to all inequalities in the system and the graph of the
linear inequality is the graph of all solutions of the system.

Example : Girlie wants to buy fruits for her mother. Her mother likes mangoes and
oranges. A mango cost 25 pesos each while and orange cost 20 pesos each. She
cannot spend more than 500 pesos but need to buy at most 15 fruits.
a. Write a system of two inequalities that describes this situation.
b. Graph the system to show all possible outcomes.
c. Is it possible for her to buy 20 mangoes and 15 oranges?

b. c. Since ( 20, 20 ) was

a. X +� ≤ 15
not located in the
25x + 20 y ≤ 500
overlapping region,
then it is not possible
to buy 20 mangoes
and 15 oranges
WHAT’S IS IT

Write your answer on a separate sheet of paper.

Write a system of equations for each problem. Then solve the system.
1. You can work at most 30 hours next week. You need to earn at least 200 pesos to cover
your weekly expenses. Your dog- walking job pays 30 per hour and your job as a car wash
attendant pays 50 per hour. Write a system of linear inequalities to model the situation.
What is the maximum number of hours he can walk the dog?

WHAT I HAVE LEARNED

Page 16 of 32
Fill in the blank to complete the instruction in every step on how to solve the system of linear
inequalities. Write your answer on a separate sheet of paper.
Step 1 Choose a different variable for each of the two ________ values you are asked to
find. Write down what each variable is to represent.
Step 2 Translate the problem into two ____________ using both variables.
Step 3 If the inequality sign does not contain an equals sign (< or >) then draw the line as
a __________.
Step 4 ________ the question or questions asked in the problem.
Step 5 Check your _________ by using the original words of the problem.

WHAT I CAN DO

Solve each problem using a system of linear inequalities Write your answer on a separate
sheet of paper.
1. David is running a concession stand at a volleyball game. He sells nachos and sodas.
Nachos cost 50 pesos each and sodas cost 20 pesos each. At the end of the game, David
made a total of at least 785 pesos and sold at least a total of 87 nachos and sodas
combined. Is it possible he had sold 100 nachos?
2. Carlos works at a movie theater selling tickets. The theater has 300 seats and charges 180
pesos for regular tickets and 140 pesos for senior citizen’s ticket. The theater expects to make
at least 20000 for each show. Is it possible that there are no senior citizens but 100 regulars
who watched?

ADDITIONAL ACTIVITY

Create a problem involving a system of linear inequalities in two variables. You can look at the
examples given previously and change the amount/hours/ numbers given. Write the system of
inequalities and solve them. Write your answer on a separate sheet of paper.

Week 3 - Day 1

What I Know

Directions: Choose the letter of the best answer. Write the chosen letter on a separate sheet of
paper.
1. It is a relation in which every element in the domain is mapped to exactly one element
in the range.
a. Function b. Relation c. One to many d. b and c
2. Some ordered pairs for a function of x are given in the table below. Which of the
following equations was used to generate the table above?

a. � = 3� − 4 b. � = 3� + 4 c. � =− 3� − 4 d. � =− 3� + 4
3. As x increases in the equation 2� + � = 4, what happens to the value of y?
a. Decreases b. Increases c. Can’t determine d. Do not
change
4. It is a set of ordered pairs.
a. Function b. Relation c. Domain d. Range
5. It is __________ if every element in the domain is mapped to a unique element in the
range.
a. Many to One b. One to many c. One to one d. Many to many

WHAT’S IN

The study on “Relations and Functions” is one of the most important topics in Algebra.
Relations and functions – these are two different words that cannot varied meaning
mathematically. You might get confused about their difference. Before we go deeper, let’s
understand the difference between both with a simple example.
In an ordered pair that is presented as (INPUT, OUTPUT): Relation shows the
relationship between INPUT and OUTPUT, whereas a function is a relation which derives one
OUTPUT for each INPUT. The illustration below shows the connection of relation and function.

Page 17 of 32
In this lesson, you will find the basics of the topic – definition of a function and a relation,
different forms of relations, the difference and similarity between relation and function.

WHAT’S NEW

Let’s start this lesson by looking at the

relationship between two things or
quantities. As you go through this lesson,
think about this question: How are the
quantities related to each other?
Description: This activity will enable you
to write ordered pairs. Out of this activity,
you can describe the relation of an object
to its common name.
Directions: Group the following objects in such a way that they have common
properties/characteristics.

WHAT IS IT

Suppose you are working in a fast food company. You earn Php 50 per hour. Your earnings are
related to the number of hours that you work.
Questions:
1. How much will you earn if you work for 4 hours a day? How about working for 5 hours
a day? 6 hours? 7 hours? Or 8 hours?
2. Express your answer as a table of values where x is the number of hours worked and y
is the total earnings.
3. Graph the table of values.
4. Suggest an equation relating x and y.

Aside from ordered pairs, a relation may be represented in three other ways: (1) table of values,
(2) graph and (3) an equation. Based on the solution, we can notice that the relation described
by the table of values show that there is exactly one value of y that corresponds to every values
of x. In mathematics, we say that y is a function of x.

WHAT’S MORE

Activity 1: REPRESENTING A RELATION

Description: Given a diagram, you will be able to learn how to make a set of ordered pairs.
Directions: Describe the mapping diagram below by writing the set of ordered pairs. The first
two coordinates are done for you.

Set of ordered pairs: {(narra, tree), (tulip, flower), (__, __), (__, __), (__, __), (__, __)}

Page 18 of 32
Activity 2: LET’S INVESTIGATE!
The weight of a person on earth and on the moon is given in the table as approximates.

1. What is the weight of a person on earth if he weighs 26 Y on the moon? 27 Y? 28 Y?

2. What is the weight of a person on earth if he weighs 174 X on the earth? 180 X? 186 X?
3. Write the set of ordered pairs using the given table.
4. Is it a relation? Why?

Activity 3: LET’S EXPLORE MORE!

A department holds a closing-out sale. All merchandise is sold at 30% discount. For the
convenience of the shoppers, the marketing supervisor considers a table of market prices (x)
and their corresponding selling prices (y). A portion of the table is given and he needs your help
to complete it.

1. Complete the given table and represents it using a mapping diagram.

2. Rewrite the table as an ordered pair then graph.
3. Determine whether the graph is a relation or a function?
4. Give the equation, describing the relation.

WHAT I HAVE LEARNED

Given the graph, complete the set of ordered pairs and table of values; draw the mapping
diagram; and generate the equation.

Based on my answer, I can say that this relation illustrates function because
_______________________.

WHAT I CAN DO

Description: This activity will enable you to make a relation, a correspondence of your
height and weight.
Directions: Form groups with 5 members each, including you. Find your height and weight
and of the other members of the group. Express the heights in centimeters and the weights in
kilograms. Write the relation of height and weight as an ordered pair in the form (height,
weight). Materials: tape measure or other measuring device, weighing device, ball pen,
paper.

ASSESSMENT

Directions: Multiple Choice. Choose the letter of the best answer. Write the chosen letter on a
separate sheet of paper.
1. Which of the following equations is a function?
A. � + �2 = 4 B. �2 + �2 = 16 C. �3 = � D. � = � + 1
2. Which of the following correspondences is a function?
A. One to one B. One to many C. Many to one D. a and c
3. It is a set of ordered pairs that contains the property that no two distinct ordered pairs have
the
same first entry.
A. Function B. Relation C. One to many D. b and
4. Given the table:

Page 19 of 32
Which of the following equations was used to generate the table above?
A. � = � − 1 B. � = � + 1 C. � =− � + 1 D. � =− � − 1
5. It is a set of ordered pairs.
A. Function B. Relation C. many to one D. b and c

Week 3 – Day 2

What I Know

Directions: Choose the letter of the best answer. Write the chosen letter on a separate sheet of
paper.
1. Which of the following equation is a relation?
A. � � = 3� + 1 B. � + �2 = 4 C. �3 = � D. � = � + 1
2. Which of the following set of ordered pairs is a function?
A. {(1,1), (2,2), (3,3), (4,4)} C. {(1,1), (1,2), (1,3), (1,4)}
B. {(1,1), (1,2), (3,3), (3,4)} D. b and c
3. Which of the following correspondence is a function?
A. One to one B. One to many C. Many to one D. a
and c
4. It is a relation in which every element in the domain is mapped to exactly one element
in the range.
A. Function B. Relation C. many to one D. b and c
5. A relation is always a function.
A. True B. Sometimes True C. Always True D. Never True

WHAT’S IN

From the previous lesson, we learned that a relation is a set of ordered pairs. But we can
also represent relation in four other ways. They are as follows: mapping diagram, graph, table
of values, equation. We also learned that function is a special type of relation. It is a set of
ordered pairs that contains the property that no two distinct ordered pairs have same first
entry.
Are you confused like the man on the picture?

This lesson will help you check if a relation is a

function? Even though a relation is represented
in different forms, in every form there are some
ways or pointers to consider whether the
relation is a function. Let’s start!

WHAT’S NEW

Since we learned from the previous lesson that relation is a set of ordered pairs and a function
is a special type of relation. Let’s see if you can identify which of these given is a relation or a
function.
Let’s start with our first activity!
Description: The aim of the activity is to determine if the relation is a function.
Direction: Tell whether the given is a relation or a function.

Page 20 of 32
 WHAT IS IT

Description: The aim of the activity is to guess the correspondence and identify which
correspondence is a relation or a function
Direction: Tell whether the given correspondence is a relation or a function.
1. How did you identify the correspondence in each mapping diagram?
2. What can you say about nos. 1 and 3? How about nos. 2
and 4?
3. Which of these mapping diagrams show relation? show a
function?
Solution: Based on the figure, there are four different
correspondences. One to one, one to many, many to one and many
to many. Two of these correspondences are both a relation then the
other two are functions. One to one and many to one
correspondence are functions, while one to many and many to
many correspondences are relations.

WHAT’S MORE
Activity 1: TOUCH ME NOT!
Description: The aim of the activity is to identify whether the graph is a relation or a function.
Direction: Lay your ball pen on the graph vertically then tell how many times the ball pen
touches the graph.

1. How many times did the ball pen touch each graph?
2. Which of following graphs is a relation? Or a function?
Solution: Based on figure, the orange line is used
for the vertical line test and the blue dot/s in the
graph shows the number of touches. If the
vertical line touches the graph once then it is a
function but if the vertical line touches the graph
more than once then it is a relation.

Activity 2
Direction: Complete the table of values below
using the given equation. Then, identify if the equation is a function.

WHAT I HAVE LEARNED

Put happy face if your answer is yes and sad face if your answer is no.

Page 21 of 32
 1. Is a function a relation?
2. If the vertical line test touches a given graph more than once, is it a function?
3. If the table of values has no same x – values, is it a function?
4. If the set of ordered pairs have no two distinct ordered pairs that have the same first
entry, is it a function?
5. If the mapping diagram shows one to one correspondence, is it a function?
6. If the equation shows that for each value of x, there is one and only one value for y,
is it a relation?

WHAT I CAN DO

Direction: Choose from the box the correct equation (rule) to show the relation between x and y
in each given.
� = �+3 � = 2� + 1
� =− 3� − 1 � = 3� + 5

ASSESSMENT

Directions: Determine whether the given is a relation or a function:

Week 3 – Day 3 & 4

What I Know

Directions: Choose the letter of the best answer. Write the chosen letter on a separate sheet of
paper.
1. In an equation where y is expressed in terms of x, the variable x is considered the
_________.
A. Dependent variable B. Independent variable C. Relation D. Ordinate
2. Given the time and salary, which of the two is the independent variable?
A. Salary B. Time C. Both D. None of the above
3. Using the given in #2, which is the dependent variable?
A. Salary B. Time C. Both D. None of the above
4. It controls the dependent variable.
A. Dependent B. Independent C. Both D. None of the above
5. In every relation there are two variables: the independent and dependent variables.
A. True B. False C. Never True D. Sometimes true

Page 22 of 32
 WHAT’S IN

From the previous lesson, we learned about relation and function. For this lesson, we are going
to learn if two variables form a relation or a function

WHAT’S NEW

Description: The aim of the activity is to determine the independent and dependent variables
in a relation.

WHAT’S IS IT

Description: Variables may either be dependent or independent. Dependent variable depends

on the independent variable while the independent variable controls the dependent variable.
Direction: Classify the variables as independent or dependent.

1. time and salary

2. the number of hours boiling and the number of liters of water in the pot
3. distance covered and the volume of gasoline
4. the number of hours studied to grade on test
5. height of a plant to the number of months grown
Generally speaking, in any given model or equation, there are two types of variables.
Independent variables are the values that can be changed or controlled in a given model or
equation. They provide the “input” which is modified by the model to change the “output”.
Dependent variables are the values that result from the independent variables. In an equation
where y is expressed in terms of x, variable x is considered the independent variable because
any value can be assigned to it. However, variable y is the dependent variable because its value
depends on the value of x.

WHAT’S MORE

Activity 1: Fill Me!

Direction: Using the previous activity, complete the following sentences to make them all true.

Activity 2: Make!
Direction: List 3 pairs of two things that are related to each other. Determine the independent
and dependent variable.

Page 23 of 32
 WHAT I HAVE LEARNED
Description: The aim of the activity is to differentiate the independent variable from the
dependent variable in a relation. Fill in the appropriate word/s to make the statement true.

ADDITIONAL ACTIVITY

Description: This task provides counter examples of independent and dependent variables.
This can be done by pair.
Direction: Each member will think of two quantities related to each other and then identify
which is the independent and dependent variables. The pair will use thumb up if they agree
with the answer of one another and thumb down if they don’t agree.

ASSESSMENT

Directions: Color the independent variable red and the dependent variable blue in each given.
1. amount of pay check and number of hours worked
2. price of speeding ticket and speed you were travelling
3. amount of rainfall and height of grass
4. speed of car and pressure applied to gas pedal
effort in class and grade in Algebra I

A. Given the mapping diagram below, write the set of ordered pairs, make a table of values;
and draw the graph.

B. Determine
whether each
relation is a
function or not.

C. Given the following experiment, identify the independent and dependent variables.
1. You want to figure out which brand of microwave popcorn pops the most kernels so you
can get the most value for your money. You test different brands of popcorn to see
which bag pops the most popcorn kernels.
2. You want to see which type of fertilizer helps plants grow the fastest, so you use a
different brand of fertilizer to each plant and see how tall they grow.
3. You’re interested in how the rising sea temperatures impact an algae’s life so you design
an experiment that measures the number of algae in a sample of water taken from a
specific ocean site under varying temperatures.

Page 24 of 32
Week 4 – Day 1, 2 & 3

What I Know

Choose the letter of the best answer. Write the chosen letter on a separate sheet of paper.

For items 1-2, given

1) What is the range of the given illustration?

A. {2, 1, 0, 1, 2} B. {0, 1, 2, 3, 4} C. {2, 1, 0} D. {1, 2, 3, 4}
2) What is the domain of the given illustration?
A. {2, 1, 0, 1, 2} B. {0, 1, 2, 3, 4} C. {2, 1, 0} D. {1, 2, 3, 4}

For items 3-4, given {(2,3),(1,-1),(0, -2), (-3, 4)}

3) What is the domain of the given ordered pairs?

A. {2, 1, 0, -3} B. {2, 1, 0, 2, -3} C. {3, -1, -2, 4} D. {3, -1, -2,
4,1}
4) What is the range of the given ordered pairs?
A. {2, 1, 0, -3} B. {2, 1, 0, 2, -3} C. {3, -1, -2, 4} D. {3, -1, -2,
4,1}
For items 5-6, given the graph,

5) What is the domain of the given graph?

A. (−∞, +∞) B. (−∞, +2) C. [−∞, +∞] D. (−∞, +2]
6) What is the range of the given graph?
A. (−∞, +∞) B. (−∞, +2) C. [−∞, +∞] D. (−∞, +2]
7). What is the domain of the function y=3x–3?
A. (−∞, -∞) B. {x/x  } C. [−∞, +∞] D. {y/y  }

For item 8-9, given the mapping diagram below.

8) What is the domain of the given illustration?
A. {0, -2, -4, -6} B. {-2, -4, -6} C. {-2, 1, 4, 7} D. {1, 4, 7}
9) What is the range of the given illustration?
A. {0, -2, -4, -6} B. {-2, -4, -6} C. {-2, 1, 4, 7} D. {1, 4, 7}
10. What is the domain of the equation y = /x-3/
A. (−∞, -∞) B. {x/x  } C. [−∞, +∞] D. {y/y  }

WHAT’S IN
Relating sets from the previous lesson, any set of ordered pairs (x, y) is called a relation in x
and y and given a relation in x and y, say “y is a function of x” if for every x, there corresponds
exactly one element y.

WHAT’S NEW
Complete the crossword below. The following terminologies were discussed from the
previous lesson.

Across
6. the set of ordered pair
7. a variable that is changed by choice
9. well defined collection of objects
10. set of first coordinates
Down
1. a relationship where every first coordinate has one and only
one second coordinate
2. the first coordinate of an ordered pair
3. set of second coordinates

Page 25 of 32
4. a comparison of terms, variables, and/or numbers
5. the second coordinate of an ordered pair
8. a variable that is changed based on another variable

WHAT IS IT

Domain
The set of first components in the ordered pairs is called the domain.
Range
The set of second components in the ordered pairs is called the range.
A. Table of Values
State the domain and range.

a) b) D: {-3, -1, 0, 2, 3)
D: {1, 2, 3)
R: (0, -1 -2, 4)
R: (2, 4, 6)
Note:

If the element of the relation appears

more than once, we write the element
once.

B. Mapping Diagram
This diagram shows correspondence between the domain ang range. The first diagram
consists the set of domains, and the second diagram consists the set of ranges.
a) b)
D: {3)
D: {100, 200, 300)
R: (1, 5)
R: (50, 100, 150)
C. Graph
The domain of a graph consists of all the input
values shown on the x-axis. The range of a graph
is the set of possible output values, which are
shown on the y-axis.

In mind, if the graph continues beyond the portion of the graph we can see, the domain and
range may be greater than the visible values.
We can observe that the graph extends horizontally from −5 to the right, without bound. So,
the domain is [−5, +∞). The vertical extent of the graph is all range values 5 and below, so the
range is (−∞,5]. Note that the domain and range are always written from smaller to larger
values, or from left to right for domain, and from the bottom to the top of the graph for range.

Examples : Find the Domain and Range given the graph:

1. 2.

Domain:{x/x∈ ℝ} or (−∞,+∞) Domain:{x/x ≥ 0}or(0, +∞}

Range: {y/y≤0} or (−∞,0) Range:{y/y∈ ℝ} or (−∞,+∞)

3. 4.

Page 26 of 32
Domain:{x/x∈ ℝ} or (−∞,+∞) Domain: {x/-3 ≤ x ≤ 3} or (-3, 3)
Range: {y/y≥-2}or(−2,+∞) Range: {y/-3 ≤ y ≤ 3} or (−3, 3)

WHAT’S MORE

Another way of finding the…

DOMAIN
 Look for input values of x which gives you an output values of y.
 Avoid 0 on the denominator of a fraction. It makes the function undefined.
 Avoid negative values under the square root sign.
RANGE
 It is the complete set of all possible resulting values of the dependent variable y, after we
have substituted the domain.
 Substitute different x-values into the expression for y to see what is happening. Ask yourself:
Is y always positive? Always negative? Maybe not equal to certain values?
 Make sure you look for minimum and maximum values of y.
 Draw a sketch! In math, it's very true that a picture is worth a thousand words.
Example:
y = x2 + 2

Solution:
There are no restrictions on the value of x. Hence, the domain is {x/x  }
or (−∞, +∞).

Since x2 is never negative, x2 + 2 is never less than 2. Hence, the range is

{y/y > 2} or [2, +∞).

Finding domain and range given the equation

A. Linear Function
The domain and range of a linear function are both(−∞,∞)

B. Absolute Value
~ If x is inside the absolute value (many to one), the
graph is V-shape opens up/down.

~ If y is inside the absolute value (one to many), the graph

is V-shape opens to the right /left

C. Equation of a Circle
~ Both x and y are squared and the same numerical
coefficient.
D. Rational Function
~ The denominator should not be equal to zero.

Illustrative Examples:

Example 1: Linear Function

Find the domain and range.
1. 2x + 3y = 6 Domain: x/x∈ ℝ} or (−∞,+∞) ; Range:{y/y∈ ℝ} or (−∞,+∞)

2. y   2 x  7 Domain: x/x∈ ℝ} or (−∞,+∞) ; Range:{y/y∈ ℝ} or (−∞,+∞)

5

Example 2: Absolute Value Equation

Find the domain and range.
a. y=|x+3| b. x=2-|2y-1|
Solution:
a. Domain = {x/x ∈R} b. Domain = {x/x ≤2}
Range = {y/y ≥0} Range = {y/y ∈R}

Page 27 of 32
Example 3: Equation of a circle
Note: In Equation of a circle, both x and y are
squared
Find the domain and range.
a. x2 + y2 = 16 b. 3x2 + 3y2 = 12

Solution:
a. √16 = 4 b. 3x2 + 3y2 = 12 ---- divide all by 3
Domain = {x/-4≤x≤4} x 2 + y2 = 4
Range = {y/-4≤y≤4} √4 = 2
Domain = {x/-2≤x≤2}
Range = {y/-2≤y≤2}

Example 4: Rational Function

Find the domain and range.
a. y  x  2 b. y  2 x  1
x3 3x  6

Solution:

a. y  x  2 the denominator x -3 ≠ 0 so x ≠ 3
x3

Domain= {x/x ∈R\3} or {x ≠ 3}

The range is not equal to the quotient of the coefficient of x in the numerator and the
cofficient
of x in the denominator. The range is not equal to the quotient of the coefficient of x in the
numerator and the cofficient of x in the denominator so y ≠ 1

Range = {y/y ∈R\1} or y ≠ 1

b. y  2 x  1 the denominator 3x – 6 ≠ 0 so x ≠ 2
3x  6
Domain = {x/x ∈R\2}
The range is not equal to the quotient of the coefficient of x in the numerator and the
cofficient
of x in the denominator so y  2
3
Range = {y/y ∈R\2/3}

WHAT I HAVE LEARNED

The domain of a function f(x) is the set of all values for which the function is defined,
and
the range of the function is the set of all values that f takes.
Domain
The set of first components in the ordered pairs is called the domain.
Range
The set of second components in the ordered pairs is called the range.

WHAT I CAN DO
A. Find the domain and range of each item.
1) 2)

B. Graph each item and find the domain and range.

3) {(-3, 4), (-2, 4), (-1, -1), (3, -1)}
4) x = -2

C. find the domain and range.

5) 6) 7) 8)

Page 28 of 32
 9) 10)

D. Find the domain and range of each function.

Equation Domain Range

1. � = � − 4
2. � = 5 � − 4
3. 2x2 + 2y2 = 50
4. 5x2 + 5y2 = 20
2�
5. � = 2�−4
6�−1
6. � = 3�−4

Week 4 – Day 4

What I Know

Directions: Write Y if the given equation represents a LINEAR FUNCTION. Write N if it is NOT
A LINEAR FUNCTION.

1. �(�) = 3� − 1
2. � � = 2�2 + 3
3. ℎ � = 5�
4. � � = 10
5. � � =� �+1
1
6. ℎ � = �−2
2
2
7. � � =
3
8. � � =4 �−6
9. ℎ � = 6� − 12
10. � � =− 12

Page 29 of 32
 WHAT’S IN

A LINEAR EQUATION is an equation in two variables which can be written in two forms:
STANDARD FORM: �� + �� = �, where �, � and � are Real Numbers and A and B are not both
0; and SLOPE-INTERCEPT FORM: � = �� + �, where � is the slope and � is the y-intercept,
where � and � are Real Numbers.

WHAT’S NEW

Suppose that your mother would like you to buy Number of

cupcakes for your sister’s birthday. The most affordable 10 20 30 40
Cupcakes
cupcakes that you found costs Php 20 per piece. Try to Cost in
200
complete the table below. Php

WHAT’S IS IT

Linear Functions
A LINEAR FUNCTION is defined by � � = �� + � , where � and � are Real Numbers.
Similar to the Slope-Intercept Form of a Linear Equation, � is the SLOPE and � is the y-
INTERCEPT.

Representations of Linear Functions

a) Equation
To determine if a given equation represents a Linear Function, we need to identify its
degree or the highest exponent of the variable in the given equation. The degree of a Linear
Function may be 0, 1 or undefined. It should also be noted that it must be written in the form
� � = �� + �.
1
Example 1: � � = 2 �3 + 7

NOT A LINEAR FUNCTION because its degree is 3.

4
Example 2: ℎ � = 5 � + 3

LINEAR FUNCTION because its degree is 1 and it is written in the form � � = �� + �.

b) Table of Values
To determine if a given Table of Values represents a Linear Function, we need to get the
first differences of the �-coordinates and the first differences of the �-coordinates or �(�) .
The differences for the � -coordinates must be equal and the differences for the � -coordinates
must also be equal.

Example 1: NOT A LINEAR FUNCTION

because the first
differences for the y-
coordinates are NOT equal.

LINEAR FUNCTION, the first

Example 2: differences for the x-coordinates
are all equal to -3 and the first
differences for the y-coordinates
are all equal to 4.

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c) Graph
The graph of a Linear Function is also a STRAIGHT LINE. Its graph may be increasing
from left to right, decreasing from left to right or it may be a horizontal line.

WHAT I HAVE LEARNED

To summarize everything that we have discussed, let use complete the following:

 To determine if an equation represents a Linear Function, we need to identify its degree

or the _______________ exponent of the variable.
 To determine if a table of values represents a Linear Function, the ________________ of the
�-coordinates must be equal and the ________________ of the �-coordinates must also be
equal.
 The graph of a Linear Function is a straight line that could be ________________,
________________ or ________________.

WHAT I CAN DO

For your TLE class, your teacher asked you to buy ribbon to decorate your project which
is a basket made of paper from a magazine. Each basket needs 32 inches of ribbon.

Is the given table representing a Linear Function? Why or why not?

ADDITIONAL ACTIVITY

In this activity, you will be exploring linear relationships among variables in real-life
situations. Choose two variables that you want to relate, then make a table of values. For example,
the table below shows the amount of mobile data that corresponds to the price of the promo
offered by a service provider. You may choose any variable that you prefer. You may gather data
from your classmates or by measuring an object’s height, length or weight. Once you have made a
table of values determine if it represents a Linear Function.

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