Math 8 Q2 Week 1-2 Module 1 Linear Inequalities in Two Variables PDF

8 Mathematics Quarter 2 – Module 1 (Week 1 & 2) Linear Inequalities in Two Variables i About the Module This module was designed and written with you in mind. It is here to help you understand about Linear Inequalities in Two Variables. The scope of this m...

8 Mathematics Quarter 2 – Module 1 (Week 1 & 2) Linear Inequalities in Two Variables i About the Module This module was designed and written with you in mind. It is here to help you understand about Linear Inequalities in Two Variables. The scope of this module permits it to be used in many different learning situations. The language used recognizes the diverse vocabulary level of students. The lessons are arranged to follow the standard sequence of the course. This module is divided into three lessons, namely:  Lesson 1 – Illustrating and Graphing Linear Inequalities in Two Variables  Lesson 2 – Problem Solving Involving System of Linear Inequalities in Two Variables After going through this module, you should be able to: 1. differentiates linear inequalities in two variables from linear equations in two variables; 2. illustrates and graphs linear inequalities in two variables; 3. solves problems involving linear inequalities in two variables; and 4. solves problems involving systems of linear inequalities in two variables. 2  What I Know (Pre-Test) Instructions: Choose the letter of the correct answer. Write your chosen answer on a separate sheet of paper. 1. Joshua bought five apples and four oranges. The total amount he paid was at most Php 200. If 𝑥 represents the cost of each apple and 𝑦 the cost of each orange, which of the following mathematical statements represents the given situation? A. 5𝑥 + 4𝑦 ≥ 200 C. 5𝑥 + 4𝑦 > 200 B. 5𝑥 + 4𝑦 ≤ 200 D. 5𝑥 + 4𝑦 < 200 2. What is the graph of linear inequalities in two variables? A. Parabola C. Half-plane B. Straight Line D. Half of a parabola 3. How many solutions does a linear inequality in two variables have? A. 0 C. 1 B. 2 D. Infinite 4. Which of the following ordered pairs is a solution of the inequality 3𝑥 + 5𝑦 ≤ 15? A. (2,2) C. (3,1) B. (3,2) D. (2,3) 5. Which of the following is a linear inequality in two variables? A. 4𝑎 − 3𝑏 = 5 C. 3𝑥 ≤ 16 B. 7𝑐 + 4 < 12 D. 𝑁𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒𝑠𝑒 2𝑥−5𝑦 ≤3 6. Which of the following graphs represents the system { ? 𝑦−3𝑥 ≤1 A. C. B. D. 3 7. Which system is not equivalent to 𝑥 + 𝑦 < 3 and 2𝑥 − 𝑦 > 1 ? 𝑥+𝑦 1 −𝑥 − 𝑦 > −3 𝑥+𝑦 1 2𝑥 + 𝑦 < 1 8. Maria makes home decorations. It takes her 2 hours to make a bouquet and 1 hour to make a basket. She can work no more than 40 hours per week. The cost to make one bouquet is Php 150 and the cost to make a basket is Php 100. She can afford to spend no more than Php 3600 per week. Write a system of inequalities to describe the situation in the problem. 2𝑥 + 𝑦 > 40 2𝑥 + 𝑦 ≤ 40 𝐴. { C. { 150𝑥 + 100𝑦 ≤ 3600 150𝑥 + 100𝑦 ≤ 3600 2𝑥 + 𝑦 ≤ 40 2𝑥 + 𝑦 < 40 𝐵. { D. { 150𝑥 + 100𝑦 < 3600 150𝑥 + 100𝑦 ≤ 3600 2𝑥 + 𝑦 < 2 9. Which of the following shows the graph of the system { ? 𝑥 − 4𝑦 > 9 A. C. B. D. 10. It is the set of points whose coordinates satisfy all the inequalities in the system. A. Solution Set B. Points C. Coordinates D. Ordered Pairs 4  Lesson ILLUSTRATING AND GRAPHING LINEAR 1 INEQUALITIES IN TWO VARIABLES What I Need To Know At the end of this lesson, you are expected to: o differentiate linear inequalities in two variables from linear equations in two variables; and o illustrate and graph linear inequalities in two variables. What’s In Can you identify the linear equations in the following examples below? 4𝑥 − 6𝑦 > 1 4𝑥 + 𝑦 = 2 4𝑥 − 3𝑦 > 4 5𝑥 − 2𝑦 = 1 6𝑥 − 4𝑦 = 7 These are all linear equations:  5𝑥 − 2𝑦 = 1  4𝑥 + 𝑦 = 2  6𝑥 − 4𝑦 = 7 A linear equation in two variables can be written in the form 𝐴𝑥 + 𝐵𝑦 = 𝐶 where a, b, c are real numbers and has ONLY one solution and order pairs. An open sentence that makes use of the symbol “=” is called an equation. On the other hand, an open sentence that makes use of any of the relation symbols , ≤, ≥, and ≠ is called an inequality. A linear inequality in two variables can be written in one of the following forms: 𝐴𝑥 + 𝐵𝑦 < 𝐶 𝐴𝑥 + 𝐵𝑦 ≤ 𝐶 𝐴𝑥 + 𝐵𝑦 > 𝐶 𝐴𝑥 + 𝐵𝑦 ≥ 𝐶 where A, B and C are real numbers and A and B are not both equal to zero. Examples: 1. 4𝑥 − 𝑦 > 1 4. 8𝑥 − 3𝑦 ≥ 14 2. 𝑥 + 5𝑦 ≤ 9 5. 2𝑦 > 𝑥 − 5 3. 3𝑥 + 7𝑦 < 2 6. 𝑦 ≤ 6𝑥 + 11 An ordered pair (x, y) is a solution of an inequality in two variables if a true statement results when the variables in the inequality are replaced by the coordinates of the ordered pair. 5  What’s New The graph of an inequality in two variables is the set of points that represents all solutions to the inequality. A linear inequality divides the coordinate plane into two halves by a boundary line where one half Plane Divider represents the solutions of the inequality. The boundary line is dashed for > and < and solid for ≤ and ≥. The half-plane that is a solution to the inequality is usually shaded. What is It To illustrate and graph linear inequalities in two variables, follow these steps: 1. Graph the boundary line of the region by rewriting the linear inequality as an equation and solving for the intercepts. 2. If the symbol of the inequality is either ≤ or ≥, draw the boundary line as a solid line. If the symbol of inequality is either < or >, draw the boundary line as a broken line. 3. Decide which half-plane contains the solution set. Pick a test point that is on one side of the boundary line. Use (0, 0) if possible. Replace x and y in the inequality with the coordinates of that point. 4. If the resulting inequality is TRUE, shade the side that contains the test point. If the resulting inequality is FALSE, shade the other side of the boundary. ILLUSTRATIVE EXAMPLE 1: Graph 2x – y ≤ 4. Solution: Let’s begin by graphing 2x – y = 4 using the intercept method. Solving for the x-intercept: y = 0 2x – 0 = 4 2x = 4 x=2 Solving for the y-intercept: (x = 0) 2(0) – y = 4 0–y=4 -1 (-y = 4) y = -4 Check point: (x = 4) 2(4) – y = 4 8–y=4 -y = 4 – 8 -1 (-y = -4) y=4 6 Since the inequality symbol ≤ includes an equal sign, the graph of 2x – y ≤ 4 includes the graph 2x – y = 4, the boundary line. Hence, when we draw the graph of 2x – y ≤ 4, we use a solid line. To check which half-plane contains the solution, substitute (0, 0) in the inequality. 2x – y ≤ 4 2(0) – 0 ≤ 4 0 ≤ 4 TRUE Hence, the half-plane that contains (0, 0) is the solution. The coordinates of every point on the shaded area and on the line 2x – y = 4 are the solutions of the inequality. Check: Let’s obtain two points from the solution set. a. (-1, 1) b. (-3, 2) 2x – y ≤ 4 2x – y ≤ 4 2(-1) – 1 ≤ 4 2(-3) – 2 ≤ 4 -2 – 1 ≤ 4 -6 – 2 ≤ 4 -3 ≤ 4 TRUE -8 ≤ 4 TRUE The points (-1, 1) and (-3, 2) satisfy the inequality. Hence, the solution is correct. ILLUSTRATIVE EXAMPLE 2: Graph the inequality 2𝑦 − 𝑥 > 6. Solution: Begin by graphing 2y – x = 6 using the intercept method. Solving for the x-intercept: y = 0 2y – x = 6 2(0) – x = 6 -1 (-x = 6) x = -6 Solving for the y – intercept: x = 0 2y – x = 6 2y – 0 = 6 2y = 6 y=3 Check point: (x = 4) 2y – 4 = 6 2y = 6 + 4 2y = 10 y=5 7  Since the inequality symbol is > , the graph of 2𝑦 − 𝑥 > 6 includes the graph 2y – x = 6, the boundary line. Hence, when we draw the graph of 2y – x = 6, we use a broken line. 2𝑦 − 𝑥 > 6 To check which half-plane contains the solution, substitute (0, 3) in the inequality. 2y – x > 6 2(3) – 0 > 6 6 > 6 FALSE Hence, the half-plane that contains (0, 3) is not part of the solution. Any point in the shaded region is a solution of the inequality 2y – x > 6 but not on the broken line. Check: Let’s obtain two points from the solution set. a. (-6, 1) b. (3, 6)) 2y – x > 6 2y – x > 6 2(1) – (-6) > 6 2(6) – 3 > 6 2+6>6 12 – 3 > 6 8 > 6 TRUE 9 > 6 TRUE The points (-6, 1) and (3, 6) satisfy the inequality. Hence, the solution is correct. Here is another way of graphing linear equations in two variables. You may follow the 3 steps below. ILLUSTRATIVE EXAMPLE 3: Graph the inequality y ≤ 2x − 1. a. The inequality already has "y" on the left and everything else on the right, so no need to rearrange. b. Plot y ≤ 2x − 1 (as a solid line c. Shade the area below because y ≤ includes equal to). (because y is less than or equal to). 8  𝑦 ILLUSTRATIVE EXAMPLE 4: Graph : 2 +2>𝑥 a. We will need to rearrange this one so "y" is on its own on the left: 𝑦 Start with: +2 > 𝑥 2 Subtract 2 from both sides: −2 −2 𝑦 >𝑥−2 2 𝑦 Multiply the inequality by 2: 2 2 (2 > 𝑥 − 2)y > 2x − 𝑦 > 2𝑥 − 4 b. Now plot 𝑦 > 2𝑥 − 4 c. Shade the area above (as a dashed line because y> does not (because y is greater than). include equals to). Note: The dashed line shows that the inequality does not include the line 𝒚 = 𝟐𝒙 − 𝟒. What’s More NOW IT’S YOUR TURN! A. Tell which of the following is a linear inequality in two variables. Write YES for linear inequality and NO for not. 1. 3𝑥 − 12 > 12 4. 9(𝑥 − 2) < 15 2. 19 > 𝑦 5. 13𝑥 + 6 ≤ 10 − 7𝑦 3. 𝑥 + 3𝑦 < 7 B. State whether each given ordered pair is a solution of the inequality. Justify your answers. 1. 2𝑥 − 𝑦 > 10; (7, 2) 2. 𝑥 + 3𝑦 ≤ 8; (4, −1) 3. 𝑦 < 4𝑥 − 5; (0, 0) 9 C. Tell which of the given coordinates of points on the graph satisfy the inequality 𝒚 < 𝟐𝒙 + 𝟐. Write YES if the given coordinates satisfy the inequality or NO if it does not. 1. (0, 2) 2. (5, 1) 𝒚 < 𝟐𝒙 + 𝟐 3. (−4, 6) 4. (8, 9) 5. (−3, −12) What I Need to Remember DON’T FORGET!  A linear inequality in two variables takes the form: y > mx + b. Linear inequalities are closely related to graphs of straight lines; recall that a straight line has the equation y = mx + b. When we graph a line in the coordinate plane, we can see that it divides the plane in half:  The solution to a linear inequality includes all the points in one half of the plane. We can tell which half by looking at the inequality sign: Inequality Solution Set Sign The solution set is the half plane above the > line. The solution set is the half plane above the line ≥ and all the points on the line. The solution set is the half plane below the < line. The solution set is the half plane below the line ≤ and all the points on the line. 10 Lesson PROBLEM SOLVING INVOLVING LINEAR 2 INEQUALITIES IN TWO VARIABLES What I Need To Know At the end of this lesson, you are expected to: o define system of linear inequalities in two variables; o pick a test point in the region to verify the solution set; and o solve problems involving linear inequalities in two variables. What’s In Let us recall all about linear inequalities and their graphs and the solution set. To graph an inequality in two variables, the following steps could be followed: 1. Replace the inequality symbol with an equal sign. The resulting equation becomes the plane divider. Examples: a. 𝑦 > 𝑥 + 6 𝑦 =𝑥+6 b. 𝑦 < −𝑥 + 2 𝑦 = −𝑥 + 2 Clipart 1 c. 𝑦 ≥ 𝑥 − 5 𝑦 =𝑥−5 d. 𝑦 ≤ −𝑥 − 3 𝑦 = −𝑥 − 3 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 < and > symbol, use a dash or a broken line. The dash or broken line indicates that the coordinates of all points on the line are not part of the solution set of the inequality. 3. Choose three points in one of the half-planes that are not on the line. Substitute the coordinates of these points into the inequality. If the coordinates of these points satisfy or make the inequality true, shade the half- plane or the region on one side of the plane divider where these points lie. Otherwise, the other side of the plane divider will be shaded. a. 𝑦 > 𝑥 + 6 b. 𝑦 < −𝑥 + 2 c. 𝑦 ≥ 𝑥 − 5 d. 𝑦 ≤ −𝑥 − 3 11  What’s New What if, there are two or more linear inequalities considered together? Example: 𝐱 – 𝟓𝐲 ≥ 𝟔 { 𝟐𝒙 + 𝒚 < 𝟑 How do call the example of two linear inequalities that are considered together? How are we going to find the solution set? Do we have separate solution set in each of the linear inequalities? Clipart 2 For us to dig deeper about the example. Let us proceed. Are you excited? What is It A system of linear inequalities in two variables consists of at least two linear inequalities or more in the same variables that are considered together. The solution of a system of linear inequalities in two variables consists of all points or ordered pairs that make all inequalities of the system true. Graphically, the coordinates of a point that lie on the graphs of all inequalities in the system is part of its solution. Clipart 3 Let’s Try! Determine whether the ordered pair is a solution of the given system of linear 𝑥 + 2𝑦 > 11 inequalities { 𝑦 ≤ 2𝑥 − 7 1. (6, 4) b. (5, 3) Solution: a. Replace x with 6 and y with 4. Since both inequalities 𝑥 + 2𝑦 > 11 𝑦 ≤ 2𝑥 − 7 are true, the ordered 6 + 2(4) > 11 4 ≤ 2(6) − 7 pair (6, 4) is a solution of the system. 6 + 8 > 11 4 ≤ 12 − 7 𝟏𝟒 > 𝟏𝟏 TRUE 𝟒 ≤ 𝟓 TRUE 2. Replace x with 5 and y with 3 Although the ordered 𝑥 + 2𝑦 > 11 𝑦 ≤ 2𝑥 − 7 pair (5, 3) satisfies the 5 + 2(3) > 11 3 ≤ 2(5) − 7 second inequality, it does not satisfy the first 5 + 6 > 11 3 ≤ 10 − 7 inequality. Thus, (5,3) is 𝟏𝟏 > 𝟏𝟏 FALSE 𝟑 ≤ 𝟑 TRUE not a solution. 12  To solve 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 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 line 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 1: To solve the system 2𝑥 – 𝑦 > −3 { graphically, graph 2𝑥 – 𝑦 > −3 and 𝑥 + 4𝑦 ≤ 9 𝑥 + 4𝑦 ≤ 9 on the same Cartesian coordinate plane. The region where the shaded regions overlap is the graph of the solution to the system. Example 2: 5𝑥 + 𝑦 > 3 To solve the system { graphically, 𝑦 ≤ 𝑥−4 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. Like systems of linear equations in two variables, systems of linear inequalities may also be applied to many real-life situations. They are used to represent situations and solve problems related to uniform motion, mixture, investment, work, and many others. Steps in solving the problem: 1. Understand the problem. 2. Write the inequalities 3. Solve the system of inequalities. 4. Check by picking up a test point to verify the solution set. Problem 1. Miya has 60 popsicle sticks. She needs to make more than 4 regular polygons using the sticks. Each polygon must be either a pentagon or a hexagon. Write 2 inequalities that represent the situations and graph. https://tinyl.io/3GVf https://tinyl.io/3GVt 13 SOLUTIONS: 1. Understand the problem. Let 𝑥 = the number of pentagons 𝑦 = the number of hexagons 2. Write the inequalities. Number of polygons must be more than 4. 𝑥+𝑦 >4 Make pentagons and hexagons with at most 60 sticks. 5𝑥 + 6𝑦 ≤ 60 3. Solve the system of inequalities. (By graphing) 𝑥+𝑦 >4 { 5𝑥 + 6𝑦 ≤ 60 a. Graph 𝑥 + 𝑦 > 4 b. Graph 5𝑥 + 6𝑦 ≤ 60 c. Combine the two graphs. solution set 4. Check: Is (5, 3) a solution? 𝑥+𝑦 >4 5+3>4 8>4 TRUE 5𝑥 + 6𝑦 ≤ 60 5(5) + 6(3) ≤ 60 25 + 18 ≤ 60 43 ≤ 60 TRUE Therefore: Yes, (5, 3) is a solution. She can make 5 pentagons and 3 hexagons (Note: Graphing calculators and graphing applications may be used.) 14 Problem 2. Alucard needs to buy at least 5 shirts. The shirt with collar costs Php 200 and shirt without collar costs Php 100. The total cost must not exceed Php 800. How many shirts of each kind can he buy? Write a system of linear inequalities and graph. SOLUTIONS: a. Understand the problem. Let 𝑥 = shirt with collar 𝑦 = shirt without collar b. Write the inequalities. Alucard needs to buy at least 5 shirts 𝑥+𝑦 ≥5 The shirt with collar costs Php 200 and shirt without collar costs Php 100. The total cost must not exceed Php 800. 200𝑥 + 100𝑦 ≤ 800 2𝑥 + 𝑦 ≤ 8 c. Solve the system of inequalities. (By graphing) 𝑥+𝑦 ≥5 { 2𝑥 + 𝑦 ≤ 8 a. Graph 𝑥 + 𝑦 ≥ 5 b. Graph 2𝑥 + 𝑦 ≤ 8 c. Combine the two graphs. solution set 15  d. Check: Is (1, 5) a solution? 𝑥+𝑦 ≥5 1+5≥5 6≥5 TRUE 2𝑥 + 𝑦 ≤ 8 2(1) + 5 ≤ 8 2+5 ≤ 8 7≤8 TRUE Therefore: Yes, (1, 5) is a solution. He can buy 1 shirt with collar and 5 shirts without collar. (Note: Graphing calculators and graphing applications may be used.) What’s More NOW IT’S YOUR TURN! A. Tell whether each ordered pair is a solution to the given system of inequalities. Write YES if it is and NO if it is not on the blank. 𝑥>2 1. { ________ a. (2, 7) ________ b. (4, 4) ________ c. (0, 0) 𝑦4 3. { ________ a. (1, 4) ________ b. (3, 5) ________ c. (8, 2) 𝑥−𝑦 4 𝑥 + 𝑦 ≥ −2 1. { 2. { 𝑥−𝑦 5 and 3𝑥 – 𝑦 ≤ 2? A. (4,-4) B. (-6, -2) C. (1, 8) D. (7, 3) 2. A cellphone producer distributes boxes of units to retail stores. A unit is either a cellphone or an accessory, and each box can have up to 24 units composed of cellphones and accessories. In addition, each box must have as any cellphones as accessories. What will be the linear inequalities to represent the situation? 𝑥 + 𝑦 < 24 𝑥 + 𝑦 ≤ 24 𝑥 + 𝑦 ≥ 24 𝑥 + 𝑦 ≤ 24 A. { B. { C. { D. { 𝑥≥𝑦 𝑥 2𝑠 7. Write the mathematical statement for Nicole bought 2 blouses (b) and 3 shirts (s) and paid not more than Php 1,150. A. 2𝑏 + 3𝑠 > 1150 C. 2𝑏 + 3𝑠 ≥ 1150 B. 2𝑏 + 3𝑠 < 1150 D. 2𝑏 + 3𝑠 ≤ 1150 8. How many solutions does a linear inequality in two variables have? A. 0 C. 1 B. 2 D. Infinite 9. What is the graph of linear inequalities in two variables? A. Parabola C. Half-plane B. Straight Line D. Half of a parabola 10. James bought three grapes and two mangoes. The total amount he paid was at most Php 150. If 𝑥 represents the cost of each grapes and 𝑦 the cost of each mangoes, which of the following mathematical statements represents the given situation? A. 3𝑥 + 2𝑦 ≥ 150 C. 3𝑥 + 2𝑦 > 150 B. 3𝑥 + 2𝑦 ≤ 150 D. 3𝑥 + 2𝑦 < 150 19  20 LESSON 2 WHAT’S MORE A. 𝑥>2 1. { a. NO b. YES c. NO 𝑦 4 3. { a. YES b. YES c. NO 𝑥−𝑦 < 6 B. 1. 2 LESSON 1 WHAT’S MORE A. B. C. 1. No 1. Yes, it is a solution. 1. No 2. No 2. Yes, it is a solution. 2. Yes 3. Yes 3. No, it is not a 3. No 4. No solution. 4. Yes 5. Yes 5. Yes HONESTY is required. Remember: This portion of the module contains all the answers. Your ANSWER KEY  21 LESSON 2 WHAT’S MORE C. 1. a. Understand the problem. Let 𝑥 = mango 𝑦 = guava b. Write the inequalities. The total number of fruits is not to exceed 8. 𝑥+𝑦≤8 A mango costs Php 20 each while a guava costs Php 15 each. He can spend no more than Php 180. 20𝑥 + 15𝑦 ≤ 180 4𝑥 + 3𝑦 ≤ 36 c. Solve the system of inequalities. (By graphing) 𝑥+𝑦 ≤8 { 4𝑥 + 3𝑦 ≤ 36 a. Graph 𝑥 + 𝑦 ≤ 8 b. Graph 4𝑥 + 3𝑦 ≤ 36 c. Combine the two graphs. d. Check: Answers may vary as long as they are located at the regions that overlapped.  22 LESSON 2 WHAT I CAN DO YES, PROBLEM SOLVED! Answers may vary. LESSON 2 WHAT’S MORE 2. a. Understand the problem. Let 𝑥 = be the number of 7-point question 𝑦 = be the number of 2-point question b. Write the inequalities. At most 45 questions 𝑥 + 𝑦 ≤ 45 Some questions are worth 7 points and the others are worth 2 points. 7𝑥 + 2𝑦 ≥ 245 Answer: The possible combinations of 7- point and 2- point questions may vary as long they are part of the solution set. References: Electronic Sources: Solving Problems involving Linear Inequalities with Two Variables, accessed September 11, 2020 https://braingenie.ck12.org/skills/106727 Solve Real World Problems Using Linear Inequalities, accessed September 20, 2020 https://students.ga.desire2learn.com/d2l/lor/viewer/viewFile.d2lfile/1798/12938 /Algebra_ReasoningwithEquationsandInequalities8.html Solving Linear Inequalities with Two Variables, accessed September 20, 2020 https://saylordotorg.github.io/text_intermediate-algebra/s05-07-solving- inequalities-with-two-.html Slope (Gradient) of a Straight Line, accessed September 2, 2020 https://www.mathsisfun.com/geometry/slope.html Books: Abuzo, Emmanuel P., et al. Mathematics 8 Learner’s Module, “Linear Inequalities in Two Variables”, Department of Education-Instructional Materials Council Secretariat. 2nd Floor Dorm G, Philsports Complex, Meralco Avenue, Pasig City. 2013 Padua, Alicia L., et.al. Our World of Math “Problems Involving Linear Inequalities”, Vibal Publishing House Inc.,1253 G. Araneta Ave., Quezon City, 2013 Abuzo, Emmanuel P., et al. Mathematics 8 Learner’s Module, “Systems of Linear Equations and Inequalities in Two Variables”, Department of Education- Instructional Materials Council Secretariat. 2nd Floor Dorm G, Philsports Complex, Meralco Avenue, Pasig City. 2013 Nivera, Gladys C. Ph.D. Grade 8 Mathematics Patterns and Practicalities, “Systems of Linear Equations and Inequalities”, Salesiana Books by Don Bosco Press Inc., Antonio Arnaiz cor. Chico Roces Avenues, Makati City, 2018 Oronce, Orlando A., et.al. E-MATH Worktext in Mathematics 8 “Systems of Linear Equations and Inequalities”, Rex Printing Company Inc., 84-86 P. Florentino St., Sta. Mesa Heights, Quezon City, 2015 Caminade, Danrie E., et al. Mathematics Workbook in Mathematics 8. “Rational Algebraic Expressions”. Department of Education – Danao City Division. Sitio Upland, National Road, Danao City. 2019 Congratulations! You are now ready for the next module. Always remember the following: 1. Make sure every answer sheet has your  Name  Grade and Section  Title of the Activity or Activity No. 2. Follow the date of submission as agreed with your teacher. 3. Keep the modules with you. 4. Return them at the end of the school year. 23  For inquiries or feedback, please write or call: Department of Education - Division of Cebu City Office Address: New Imus Avenue, Barangay Day-as, Cebu City Telephone Nos: (032) 255-1516/ (032) 328-8899 Email Address: [email protected] 24