8

Mathematics
Second Quarter
Module 2: Graphing Linear
Inequalities in Two Variables

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 8

Mathematics
Second Quarter
Module 2: Graphing Linear
Inequalities in Two Variables

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 INTRODUCTION

This module is written in support of the K to 12 Basic Education Program to ensure

attainment of standards expected of you as a learner.
This aims to equip you with essential knowledge on Illustrating and Graphing
Linear Inequalities in Two Variables.
This includes the following activities/tasks:
 Expected Learning Outcome – This lays out the learning outcome that you are
expected to have accomplished at the end of the module.
 Pre-test – This determines your prior learning on the particular lesson you are
about to take.
 Discussion of the Lesson – This provides you with the important knowledge,
principles and attitudes that will help you meet the expected learning
outcome.
 Learning Activities – These provide you with the application of the knowledge
and principles you have gained from the lesson and enable you to further
enhance your skills as you carry out prescribed tasks.
 Post-test – This evaluates your overall understanding about the module.

With the different activities provided in this module, may you find this material
engaging and challenging as it develops your critical thinking skills.

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 What I Need to Know

At the end of this lesson, you will be able to:

 illustrate and graph linear inequalities in two variables.

What I Know

I. State whether each given ordered pair is a solution of the inequality. Write S if it is
a solution and NS if it is not.
1) 2x – y > 10; (7, 2)
2) -3x + y < -12; (0, -5)
3) x + 3y ≤ 8; (4, -1)
4) 9 + x ≥ y; (-6, 3)
5) y < 4x – 5; (0, 0)
6) 2y – 2x ≤ 14; (-3, -3)
7) 7x – 2y ≥ 6; (-3, -8)
8) 16 – y > x; (-1, 9)

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II. Find the graph of the linear inequalities from the given set of graphs below. Write
the letter of your choice on your notebook.
9) y > 4x – 4
10) 3y < 5x + 2
11) 4y ≥ -3x + 4
12) 2y ≤ 5x – 4

a. b.

c. d.

What`s In

Directions: Show the graph of each of the following linear equations in a Cartesian
coordinate plane. Do it on a graphing paper.

1. y = -2x + 2
2. 2y = x + 3

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 What`s New

Below is the graph of the linear equation y = x + 3. Use the graph to answer
the following questions.

Questions:
1. How would you describe the line in relation to the plane where it lies?
2. Name five points on the line y = x + 3. What can you say about the
coordinates of these points?
3. Name five points not on the line y = x + 3. What can you say about the
coordinates of these points?
4. What mathematical statement would describe all the points on the left side of
the line y = x + 3?
5. How about all the points on the right side of the line y = x + 3?
6. What conclusion can you make about the coordinates of points on the line
and those which are not on the line?

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 What Is It

Read and Learn!!!

In algebra, the graph of an equation is a representation of its solution. The
solution of a linear equation in two variables is represented by all points that are on
its line.
The same is true with the graph of an inequality. The solution of a linear
inequality in two variables is represented by a shaded region, or area. This implies
that all points in the shaded area are solutions in the inequality.
Graphs of Linear Inequalities in Two Variables
Graphing linear inequalities in two variables builds on your skill of graphing
linear equations.
When a line is graphed in the coordinate plane, it separates the plane into two
regions called half- planes. The line that separates the plane is called the plane
divider.
The lines of graphing linear inequalities in two variables are as follows:
 If the linear inequality is < or >, use broken line. The broken line
indicates that all points on the line are not part of the solution set of the
inequality.
 If the linear inequality is ≤ or ≥, use solid line. The solid line indicates
that all points on the line are part of the solution of the inequality.

Steps on Graphing Linear Inequalities in Two variables

1. The first step in the process is graphing a line using any of the methods
you’ve learned; slope and y-intercept, intercepts, two points, or table of
values. To do this, change the inequality symbol to equal sign “=”.
2. The next step in the process is using a test point to determine which side to
shade. A test point can be any point on either side of the line graph. To do
this, change the equal sign to its inequality symbol. Substitute the values of x
and y in the inequality.
3. Finally, if the test point is a solution, shade the whole area where the test
point is. If it is not a solution, shade the opposite side where the test point is.

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Example 1: Graph
Step 1: Graph the line
 To do this, change the inequality symbol to equal sign.
5x + 2y = 14
 Graph using intercepts

Find the x-intercept.

5x + 2y = 14
5x + 2(0) = 14
5x = 14
x = 14/5

Find the y-intercept

5x + 2y = 14
5(0) + 2y = 14
2y = 14
y=7

 Plot the x-intercept 14/5 on the x-axis and y-intercept 7 on the y-axis. Graph
the line by drawing a solid line connecting the two points.

Step 2: Determine the shaded area by using a test point.

 The easiest test point is (0,0), the origin. Substitute x = 0 and y = 0 to the
inequality 5x + 2y ≥ 14.
5(0) + 2(0) ≥ 14
0 ≥ 14

 Since 0 is not greater than 14, this statement is false. This implies that the
side of the line where the origin is found (left side of the line) does not contain
all the possible solutions of the inequality. Instead, all the possible solutions of
the inequality lie on the line and its right side of the line.

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Step 3: Shade the area containing the solution.

Example 2: Graph .
Step 1: Graph the line
 To do this, change the inequality symbol to equal sign.
-2x + y = 2
 Graph using intercepts

Find the x-intercept.

-2x + 0 = 2
-2x = 2
x = -1

Find the y-intercept

-2x + y = 2
-2(0) + y = 2
y=2

 Plot the x-intercept -1 on the x-axis and y-intercept 2 on the y-axis. Graph the
line by drawing a solid line connecting the two points.

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Step 2: Determine the shaded area by using a test point.
 The easiest test point is (0,0), the origin. Substitute x = 0 and y = 0 to the
inequality -2x + y ≤ 2.
-2(0) + 0 ≤ 2
0≤2

 Since 0 is less than 2, this statement is true. This implies that the side of the
line where the origin is found (right side of the line) contains all the possible
solutions of the inequality. Instead, all the possible solutions of the inequality
lie on the line and its right side of the line.

Step 3: Shade the area containing the solution.

Example 3: Graph
Step 1: Graph the line
 To do this, change the inequality symbol to equal sign.
3x + y = -3

 Graph using intercepts

Find the x-intercept.

3x + y = -3
3x + 0 = -3
3x = -3
x = -1

Find the y-intercept

3x + y = -3
3(0) + y = -3
y = -3

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  Plot the x-intercept -1 on the x-axis and y-intercept -3 on the y-axis. Graph the
line by drawing broken lines connecting the two points.

Step 2: Determine the shaded area by using a test point.

 The easiest test point is (0,0), the origin. Substitute x = 0 and y = 0 to the
inequality 3x + y > -3.
3(0) + 0 > -3
0 > -3

 Since 0 is greater than -3, this statement is true. This implies that the side of
the line where the origin is found (right side of the line) contains all the
possible solutions of the inequality. Instead, all the possible solutions of the
inequality lie on its right side of the line.

Step 3: Shade the area containing the solution.

Example 4: Graph .
Step 1: Graph the line
 To do this, change the inequality symbol to equal sign.
4x + y = 4

 Graph using intercepts

Find the x-intercept.

4x + y = 4
4x + 0 = 4
4x = 4
x=1

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 Find the y-intercept
4x + y = 4
4(0) + y = 4
y=4

 Plot the x-intercept 1 on the x-axis and y-intercept 4 on the y-axis. Graph the
line by drawing broken lines connecting the two points.

Step 2: Determine the shaded area by using a test point.

 The easiest test point is (0,0), the origin. Substitute x = 0 and y = 0 to the
inequality 4x + y < 4.
4(0) + 0 < 4
0<4

 Since 0 is less than 4, this statement is true. This implies that the side of the
line where the origin is found (left side of the line) contains all the possible
solutions of the inequality. Instead, all the possible solutions of the inequality
lie on its left side of the line.

Step 3: Shade the area containing the solution.

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 What`s More

Independent Activity
Task: Graph y > x – 2 in a Cartesian Coordinate Plane.

Independent Assessment
Directions: Based on your graph, tell whether the given ordered pair is a solution or
not. Write solution if it is one of the solutions and not solution if it is not.

1. (1, -1)
2. (4, 0)
3. (2, 3)
4. (0, 5)
5. (-2, 8)
6. (0, 2)
7. (5, 1)
8. (-4, 6)

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 What I Have Learned

I learned that:
 when a line is graphed in the coordinate plane, it separates the plane into two
regions called half- planes. The line that separates the plane is called the
plane divider.

 the lines of graphing linear inequalities in two variables are as follows:

 If the linear inequality is < or >, use broken line. The broken line
indicates that all points on the line are not part of the solution set of the
inequality.
 If the linear inequality is ≤ or ≥, use solid line. The solid line indicates
that all points on the line are part of the solution of the inequality.

 the steps in graphing linear inequality in two variables.

 The first step in the process is graphing a line using any of the methods
you’ve learned; slope and y-intercept, intercepts, two points, or table of
values. To do this, change the inequality symbol to equal sign “=”.
 The next step in the process is using a test point to determine which
side to shade. A test point can be any point on either side of the line
graph. To do this, change the equal sign to its inequality symbol.
Substitute the values of x and y in the inequality.
 Finally, if the test point is a solution, shade the whole area where the
test point is. If it is not a solution, shade the opposite side where the
test point is.

What I Can Do

Directions: Graph the following linear inequality in two variables using the steps
you’ve learned. Do this on a graphing paper.
1)
2)

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 Assessment

I. Tell whether the given ordered pair is a solution of the inequality shown in the
graph. Write S if it is a solution and NS if it is not.

1)
2)
3)
4)
5)
6)
7)
8)

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II. Find the graph of the linear inequalities on the illustrations below. Write only the
letter of your choice on your notebook.
1)
2)
3)
4)

a. b

c. d.

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 References

Callanta, Melvin M., Allan M. Canonigo, Arnaldo I. Chua, Jerry D. Cruz,

Mirla S. Esparrago, Elino S. Garcia, Aries N. Magnaye, Fernando Orines,
Rowena S. Perez, and Concepcion S. Ternida, eds. Mathematics Learner’s
Module Grade 8. First Edition, Pasig City.
Rex Book Store, Inc. 2015
Diaz, Zenaida B, Maharlika P. Mojica, Catalina B. Manalo, Josephine L.
Suzara, Jesus P. Mercado, Mirla S. Esparrago, Nestor V. Reyes, Jr. and
Fernando B. Orines. Next Century Mathematics. Phoenix Publishing House.

https://www.slideshare.net/rafullido/math-8-linear-inequalities

https://www.google.com/search?q=graph+of+linear+inequalities+in+two+variables&t
bm=isch&ved=2ahUKEwjHpYmEwtHrAhUWH3IKHT-xBQgQ2-
cCegQIABAA&oq=graph+of+linear+inequalities+in+two+variables&gs_lcp=CgNpbW
cQDDICCAA6BAgAEB

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