Module 2: Systems of Linear Inequalities in Two Variables

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Republic of the Philippines

Department of Education Regional Office IX, Zamboanga Peninsula

Mathematics Quarter 2 - Module 2:

Systems of Linear Inequalities in Two Variables

Zest for Progress

Zeal of Partnership

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Name of Learner: ___________________________

Grade & Section: ___________________________

Name of School: ___________________________

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Systems of Linear Inequalities in Two Variables What I Need to Know

Objective: At the end of this module the learner will be able to solve problems involving systems of linear inequalities in two variables. (M8AL-IIb-2).

What’s In Activity 1: GRAPH IT

Direction: Graph the inequalities :

1. 5x - y > 3 2. 4 xy

Questions:

1. How did you graph each mathematical statement?

2. How many solutions does a linear inequality in two variable have?

3. Suppose you drew the graphs of 5x - y > 3and y ≤ x + 4 in the same Cartesian coordinate plane. How would you describe their graph?

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What’s New Activity 2: LOOK AT ME

Study the graph below and answer the questions that follow.

5x + y > 3

y≤ x - 4

Questions:

1. What have you notice about the graph of the two inequalities?

2. Are their shaded areas overlap?

3. In what graphs of the inequalities does point A lie? How about point B?

4. Can you find any point that lie on the graphs of all the inequalities? If yes, do these points satisfy to both inequalities?

5. What can you say about the points that lie on the graphs of all

the inequalities?

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What is it An ordered pair (x,y) is a solution to a system of inequalities if it satisfies all the inequalities

in the system. Graphically, the coordinates of a point that lie on the graphs of all inequalities in the system is part of its solution.

To graph a system of inequalities in two variables by graphing.

1. Draw the graph of each inequality on the same coordinate plane. Shade the appropriate half-plane. Recall that if all points on the same line are included on the line are included in the solution, it is a closed half-plane, and the line is solid. On the other hand, if the points on the lines are not part of the solution of the inequality, it is an open half-plane and the line is broken.

2. The region where shaded areas overlap is the graphical solution to the system. If the graphs do not overlap, then the system has no solution.

Example: To solve the system graphically, graph 5x + y > 3 and y ≤ x - 4 on the same

Cartesian coordinate plane. The region where the shaded regions overlap is the graph of the

solution to the system.

5x +y>3

This shaded part is the solution to the system.

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35

xy

yx

4 xy

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Test the some points.

Questions:

1. Which point makes either of the inequalities true? Is it part of the solution set?

2. How about the point which makes both the inequalities true? Is it part of the solution set?

3. When can a point be a solution of the system?

4. How many points can be a solution of the system?

Point A (5,-2)

Substitute the coordinates in the inequalities

5x + y >3

x = 5 and y = -2

5(5) + (-2) >3

23 > 3 , True

y ≤ x- 4

-2 ≤ 5 - 4

-2 ≤ 1, True

Therefore point A(5,-2) is a solution set of the inequalities.

Point B (2,3)

Substitute the coordinates in the inequalities

5x + y >3

x = 2 and y = 3

5(2) + 3 >3

13 >3, True

y ≤ x- 4

3 ≤ 2 - 4

3 ≤ - 2, False

Therefore point B(2,3) is not a solution set of the inequalities.

Point C (1, -4)

Substitute the coordinates in the inequalities

5x + y >3

x =1 and y = -4

5(1) + (-4) > 3

1 > 3, False

y ≤ x- 4

-4 ≤ 1 - 4

-4 ≤ -3, True

Therefore point C (1,-4) is not a solution set of the inequalities.

Point D (2,-3)

Substitute the coordinates in the inequalities

5x + y >3

x = 2 and y = -3

5(2) +(-3)> 3

7 > 3, True

y ≤ x- 4

-3 ≤ 2 - 4

-3 ≤ -2, True

Therefore point D (2,-3) is a solution set of the inequalities.

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What’s more

Activity 3: DO I SATISFY YOU?

Direction: Determine if each ordered pair is a solution of the system of the linear inequality .

42

5

yx

yx Then, answer the questions that follow.

1. (3,-1)

2. (4,-2)

3. (3,1)

4. (3,-6)

5. (5,-3)

6. (2,-15)

7. (-5,2)

8. (0, 1)

9. (-1, 2)

10. (1, 12)

A. How did you determine if the given ordered pair is a solution of the system?

B. How did you know that the given ordered pair is not a solution of the system?

x + y < 5

2x-y >4

6

14

92

xy

xy

53

12

xy

yx

Activity 4: AM I THAT REGION?

Directions : Solve the following systems of inequalities graphically. Find three points that satisfy both inequalities. Plot the points to show that they belong to the solution of the system.

1. 2. 93

93

yx

yx

3.

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What I Have Learned Activity 5: REGION IN A PLANE

Study the graph and answer the questions below.

3x + 2y ≤ 12

X - 2y > 4

1. Which shaded part is the solution set?____________________________________

____________________________________________________________________

2. What point is part of the solutions? _____________

3. Are the points on the line x - 2y = 4 included in the solution set? Explain your answer.__________________________________________________________________________________________________________________________________

4. How about the points on the line 3x +2y = 12? Explain your answer.

____________________________________________________________________

5. Identify at least three points as part of the solution of the system.

_________, _________, _________.

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What I can Do Activity 6: SHADE ME!

Direction: Shade the side of the plane divider where the solutions of the inequalities are found.

1. 623

1223

yx

yx 2.

623

1223

yx

yx

3. 623

1223

yx

yx

4. 623

1223

yx

yx

Answer the following questions?

1. How do you determine the solution set of a system of linear inequalities in two variables

from its graph?

2. Do you think it is easy to determine the solution set of a system of linear inequalities by

graphing? Explain your answer.

3. Which system of linear inequalities has no solution? Why?

4. When can you say that a system of linear inequalities has a solution? No solution?

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Assessment: Direction: . Choose the letter that you think best answers the question.

1. Which of the following is a system of linear equations in two variables?

A. 2x – 7y = 8 C. 9 2

2 3 12

B. 3 5 2

4 9 D. 4x + 1 = 8

2. What is the first step in graphing the solution set of an inequality?

A. Graph each equation C. Test a point

B. Identify the unknown D. Shade the half-plane

3. If the point satisfies the inequality, what does it mean?

A. It is part of the solution set. A. It is not part of the solution set. B. It is on the plane divider. C. It doesn’t mean at all.

4. What point is the intersection of the graphs of the lines x + y = 8 and 2x – y = 1 ?

A. (1, 8 ) B. ( 3, 5 ) C. ( 5, 3 ) D( 2, 6 )

5. Which of the following is a graph of a system of linear inequalities in two variables?

A. C.

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B. D.

6. Which of the following shows the graph of the system 2 2

4 9 ?

A. C.

B. D.

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7. Which of the following ordered pairs satisfy both 2x + 7y > 5 and 3x – y ≤ 2 ?

A. ( 0, 0) B. (10, -1) C. (-4, 6) D. (-2, -8)

8. Graph the solution set of 4

2

yx

yx

A. C.

B. D.

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9. The graph of the system of inequalities 3

02

xy

y

A. C.

B. D.

10. A solution of a system of linear inequalities is an order pair that

A. is a solution of both inequalities.

B. Is a solution in one inequality.

C. Is a solution in either of the inequalities.

D. Is a solution in neither of the inequalities.

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Writer: ZITA A. ZAYAS REVIEWER: ISMAEL K. YUSOP Management Team: SDS : Ma. Liza R. Tabilon EdD,CESO V ASDS : Dr. Judith V. Romaguera OIC- ASDS : Dr. Judelyn J. Ramos OIC- ASDS : Mr. Armando P. Gumapon CID Chief : Dr. Lilia E. Abello LR: Dr. Evelyn C. Labad PSDS : Dr. Glenda B. Gudmalin Principal : Cristina D. Gumapon

References Emmanuel P. Abuzo, Merden L. Bryant, Jem Boy b. Cabrella, Belen P. Caldez, Melvin M. Callanta, Anastacia Proserfina I. Castro, Alicia R. Halabaso, Sonia P. Javier, Roger T. Nocom, and Concepcion S. Ternida, Mathematics- Grade 8 Learner’s Module First edition, 2013. pp 243-298, August 2020

Fernando B. Orines , Zenaida B. Diaz, Maharlika P. Mojica, Catalina B. Manalo, Josephine L. Suzara, Jesus P. Mercado, Mirla S. Esparrago, Nestor V. Reyae Jr., Copyright 2013. Next Century Mathematics 8, pp 112-114, August 2020.

www.geogebra.org, Geogebra Classic application

Mathematics - Grade 8 Alternative Delivery Mode Quarter 2 - Module 2 : Systems of Linear Inequalities in two variables First Edition, 2020

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Published by the Department of Education - Division of Zamboanga del Norte Secretary: Leonor Magtolis Briones

Printed in the Philippines by Department of Education – Region IX Zamboanga Peninsula Office Address: Pob. North, Pinan, Zamboanga del Norte Telefax: (062) 215 3747 E-mail Address: [email protected]

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