Linear Inequalities in Two Variables PDF

LINEAR INEQUALITIES 4 IN TWO VARIABLES I. INTRODUCTION AND FOCUS QUESTIONS Have you asked yourself how your parents budget their income for your family’s needs? How engineers determine the needed materials in the construction of new houses, b...

LINEAR INEQUALITIES 4 IN TWO VARIABLES I. INTRODUCTION AND FOCUS QUESTIONS Have you asked yourself how your parents budget their income for your family’s needs? How engineers determine the needed materials in the construction of new houses, bridges, and other structures? How students like you spend their time studying, accomplishing school requirements, surfing the internet, or doing household chores? These are some of the questions which you can answer once you understand the key concepts of Linear Inequalities in Two Variables. Moreover, you’ll find out how these mathematics concepts are used in solving real-life problems. II. LESSONS AND COVERAGE In this module, you will examine the above questions when you take the following lessons: Mathematical Expressions and Equations in Two Variables Equations and Inequalities in Two Variables Graphs of Linear Inequalities in Two Variables 193  In these lessons, you will learn to: differentiate between mathematical expressions and mathematical equations; differentiate between mathematical equations and inequalities; illustrate linear inequalities in two variables; graph linear inequalities in two variables on the coordinate plane; and solve real-life problems involving linear inequalities in two variables. Module Map Module Map This chart shows the lessons that will be covered in this module. Mathematical Expressions and Equations in Two Variables Equations and Inequalities in Two Linear Inequalities in Two Variables Variables Graphs of Linear Inequalities in Two Variables 194 III. PRE - ASSESSMENT Find out how much you already know about this module. Choose the letter that corresponds to your answer. Take note of the items that you were not able to answer correctly. Find the right answer as you go through this module. 1. Janel bought three apples and two oranges. The total amount she paid was at most Php 123. If x represents the number of apples and y the number of oranges, which of the following mathematical statements represents the given situation? a. 3x + 2y ≥ 123 c. 3x + 2y > 123 b. 3x + 2y ≤ 123 d. 3x + 2y < 123 2. How many solutions does a linear inequality in two variables have? a. 0 b. 1 c. 2 d. Infinite 3. Adeth has some Php 10 and Php 5 coins. The total amount of these coins is at most Php 750. Suppose there are 50 Php 5-coins. Which of the following is true about the number of Php 10-coins? I. The number of Php 10-coins is less than the number of Php 5-coins. II. The number of Php 10-coins is more than the number of Php 5-coins. III. The number of Php 10-coins is equal to the number of Php 5-coins. a. I and II b. I and III c. II and III d. I, II, and III 4. Which of the following ordered pairs is a solution of the inequality 2x + 6y ≤ 10? a. (3, 1) b. (2, 2) c. (1, 2) d. (1, 0) 5. What is the graph of linear inequalities in two variables? a. Straight line c. Half-plane b. Parabola d. Half of a parabola 6. The difference between the scores of Connie and Minnie in the test is not more than 6 points. Suppose Connie’s score is 32 points, what could be the score of Minnie? a. 26 to 38 b. 38 and above c. 26 and below\ d. between 26 and 38 195  7. What linear inequality is represented by the graph at the right? a. x–y>1 b. x–y 1 d. -x + y < 1 8. In the inequality c – 4d ≤ 10, what could be the possible value of d if c = 8? 1 1 1 1 a. d≤- b. d≥- c. d≤ d. d≥ 2 2 2 2 9. Mary and Rose ought to buy some chocolates and candies. Mary paid Php 198 for 6 bars of chocolates and 12 pieces of candies. Rose bought the same kinds of chocolates and candies but only paid less than Php 100. Suppose each piece of candy costs Php 4, how many bars of chocolates and pieces of candies could Rose have bought? a. 4 bars of chocolates and 2 pieces of candies b. 3 bars of chocolates and 8 pieces of candies c. 3 bars of chocolates and 6 pieces of candies d. 4 bars of chocolates and 4 pieces of candies 10. Which of the following is a linear inequality in two variables? a. 4a – 3b = 5 c. 3x ≤ 16 b. 7c + 4 < 12 d. 11 + 2t ≥ 3s 11. There are at most 25 large and small tables that are placed inside a function room for at least 100 guests. Suppose only 6 people can be seated around the large table and only 4 people for the small tables. How many tables are placed inside the function room? a. 10 large tables and 9 small tables b. 8 large tables and 10 small tables c. 10 large tables and 12 small tables d. 6 large tables and 15 small tables 196 12. Which of the following shows the plane divider of the graph of y ≥ x + 4? a. c. b. d. 13. Cristina is using two mobile networks to make phone calls. One network charges her Php 5.50 for every minute of call to other networks. The other network charges her Php 6 for every minute of call to other networks. In a month, she spends at least Php 300 for these calls. Suppose she wants to model the total costs of her mobile calls to other networks using a mathematical statement. Which of the following mathematical statements could it be? a. 5.50x + 6y = 300 c. 5.50x + 6y ≥ 300 b. 5.50x + 6y > 300 d. 5.50x + 6y ≤ 300 14. Mrs. Roxas gave the cashier Php 500-bill for 3 adult’s tickets and 5 children’s tickets that cost more than Php 400. Suppose an adult ticket costs Php 75. Which of the following could be the cost of a children’s ticket? a. Php 60 b. Php 45 c. Php 35 d. Php 30 197 15. Mrs. Gregorio would like to minimize their monthly bills on electric and water consumption by oberving some energy and water saving measures. Which of the following should she prepare to come up with these energy and water saving measures? I. Budget Plan II. Previous Electric and Water Bills III. Current Electric Power and Water Consumption Rates a. I and II b. I and III c. II and III d. I, II, and III 16. The total amount Cora paid for 2 kilos of beef and 3 kilos of fish is less than Php 700. Suppose a kilo of beef costs Php 250. What could be the maximum cost of a kilo of fish to the nearest pesos? a. Php 60 b. Php 65 c. Php 66 d. Php 67 17. Mr. Cruz asked his worker to prepare a rectangular picture frame such that its perimeter is at most 26 in. Which of the following could be the sketch of a frame that his worker may prepare? a. c. b. d. 198 18. The Mathematics Club of Masagana National High School is raising at least Php 12,000 for their future activities. Its members are selling pad papers and pens to their school- mates. To determine the income that they generate, the treasurer of the club was asked to prepare an interactive graph which shows the costs of the pad papers and pens sold. Which of the following sketches of the interactive graph the treasurer may present? a. c. b. d. 19. A restaurant owner would like to make a model which he can use as guide in writing a linear inequality in two variables. He will use the inequality in determining the number of kilograms of pork and beef that he needs to purchase daily given a certain amount of money (C), the cost (A) of a kilo of pork, the cost (B) of a kilo of beef. Which of the following models should he make and follow? I. Ax + By ≤ C II. Ax + By = C III. Ax + By ≥ C a. I and II b. I and III c. II and III d. I, II, and III 20. Mr. Silang would like to use one side of the concrete fence for the rectangular pig pen that he will be constructing. This is to minimize the construction materials to be used. To help him determine the amount of construction materials needed for the other three sides whose total length is at most 20 m, he drew a sketch of the pig pen. Which of the following could be the sketch of the pig pen that Mr. Silang had drawn? a. c. b. d. 199 What to What to Know Know Start the module by assessing your knowledge of the different mathematical concepts previously studied and your skills in performing mathematical operations. This may help you in understanding Linear Inequalities in Two Variables. As you go through this module, think of the following important question: “How do linear inequalities in two variables help you solve problems in daily life?” To find out the answer, perform each activity. If you find any difficulty in answering the exercises, seek the assistance of your teacher or peers or refer to the modules you have gone over earlier. To check your work, refer to the answers key provided at the end of this module. A ctivity 1 WHEN DOES LESS BECOME MORE? Directions: Supply each phrase with the most appropriate word. Explain your answer briefly. 1. Less money, more __________ 2. More profit, less __________ 3. More smile, less __________ 4. Less make-up, more __________ 5. More peaceful, less __________ 6. Less talk, more __________ 7. More harvest, less __________ 8. Less work, more __________ 9. Less trees, more __________ 10. More savings, less __________ ES TIO a. How did you come up with your answer? ? QU NS b. How did you know that the words are appropriate for the given phrases? c. When do we use the word “less”? How about “more”? d. When does less really become more? e. How do you differentiate the meaning of “less” and “less than”? How are these terms used in Mathematics? 200  f. How do you differentiate the meaning of “more” and “more than”? How are these terms used in Mathematics? g. Give at least two statements using “less”, “less than”, “more” and “more than”. h. What other terms are similar to the terms “less”, “less than”, “more” or “more than”? Give statements that make use of these terms. i. In what real-life situations are the terms such as “less than” and “more than” used? How did you find the activity? Were you able to give real-life situations that make use of the terms less than and more than? In the next activity, you will see how inequalities are illustrated in real-life. A ctivity 2 BUDGET…, MATTERS! Directions: Use the situation below to answer the questions that follow. Amelia was given by her mother Php 320 to buy some food ingredients for “chicken adobo”. She made sure that it is good for 5 people. E S TI O 1. Suppose you were Amelia. Complete the following table with the ? needed data. NS QU Cost per unit Estimated Ingredients Quantity or piece Cost chicken soy sauce vinegar garlic onion black pepper sugar tomato green pepper potato 201  2. How did you estimate the cost of each ingredient? 3. Was the money given to you enough to buy all the ingredients? Justify your answer. 4. Suppose you do not know yet the cost per piece or unit of each ingredient. How will you represent this algebraically? 5. Suppose there are two items that you still need to buy. What mathematical statement would represent the total cost of the two items? From the activity done, have you seen how linear inequalities in two variables are illustrated in real life? In the next activity, you will see the differences between mathematical expressions, linear equations, and inequalities. A ctivity 3 EXPRESS YOURSELF! Directions: Shown below are two sets of mathematical statements. Use these to answer the questions that follow. y = 2x + 1 y > 2x + 1 3x + 4y = 15 10 – 5y = 7x 3x + 4y < 15 10 – 5y ≥ 7x y = 6x + 12 9y – 8 = 4x y ≤ 6x + 12 9y – 8 < 4x ES TIO 1. 2. How do you describe the mathematical statements in each set? What do you call the left member and the right member of each ? QU NS mathematical statement? 3. How do you differentiate 2x + 1 from y = 2x + 1? How about 9y – 8 and 9y – 8 = 4x? 4. How would you differentiate mathematical expressions from mathematical equations? 5. Give at least three examples of mathematical expressions and mathematical equations. 6. Compare the two sets of mathematical statements. What statements can you make? 7. Which of the given sets is the set of mathematical equations? How about the set of inequalities? 8. How do you differentiate mathematical equations from inequalities? 9. Give at least three examples of mathematical equations and inequalities. 202  Were you able to differentiate between mathematical expressions and mathematical equations? How about mathematical equations and inequalities? In the next activity, you will identify real-life situations involving linear inequalities. A ctivity 4 “WHAT AM I?” Directions: Identify the situations which illustrate inequalities. Then write the inequality model in the appropriate column. Classification Real-Life Situations Inequality Model (Inequality or Not) 1. The value of one Philippine peso (p) is less than the value of one US dollar (d). 2. According to the NSO, there are more female (f) Filipinos than male (m) Filipinos. 3. The number of girls (g) in the band is one more than twice the number of boys (b). 4. The school bus has a maximum seating capacity (c) of 80 persons 5. According to research, an average adult generates about 4 kg of waste daily (w). 6. To get a passing mark in school, a student must have a grade (g) of at least 75. 7. The daily school allowance of Jillean (j) is less than the daily school allowance of Gwyneth (g). 8. Seven times the number of male teachers (m) is the number of female teachers (f). 9. The expenses for food (f) is greater than the expenses for clothing (c). 10. The population (p) of the Philippines is about 103 000 000. 203  ES TIO 1. How do you describe the situations in 3, 5, 8 and 10? How about the ? QU NS situations in 1, 2, 4, 6, 7 and 9? 2. How do the situations in 3, 5, 8 and 10 differ from the situations in 1, 2, 4, 6, 7 and 9? 3. What makes linear inequality different from linear equations? 4. How can you use equations and inequalities in solving real-life problems? From the activity done, you have seen real-life situations involving linear inequalities in two variables. In the next activity, you will show the graphs of linear equations in two variables. You need this skill to learn about the graphs of linear inequalities in two variables. A ctivity 5 GRAPH IT! A RECALL… Directions: Show the graph of each of the following linear equations in a Cartesian coordinate plane. 1. y = x + 4 2. y = 3x – 1 3. 2x + y = 9 4. 10 – y = 4x 5. y = -4x + 9 204  ES TIO 1. How did you graph the linear equations in two variables? ? QU NS 2. How do you describe the graphs of linear equations in two variables? 3. What is the y-intercept of the graph of each equation? How about the slope? 4. How would you draw the graph of linear equations given the y-intercept and the slope? Were you able to draw and describe the graphs of linear equations in two variables? In the next task, you will identify the different points and their coordinates on the Cartesian plane. These are some of the skills you need to understand linear inequalities in two variables and their graphs. A ctivity 6 INFINITE POINTS……… Directions: Below is the graph of the linear equation y = x + 3. Use the graph to answer the following questions. ES TIO 1. How would you describe the line in relation to the plane where it lies? 2. Name 5 points on the line y = x + 3. What can you say about the ? QU NS coordinates of these points? 3. Name 5 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? How about all the points on the right side of the line y = x + 3? 5. What conclusion can you make about the coordinates of points on the line and those which are not on the line? 205  From the activity done, you were able to identify the solutions of linear equations and linear inequalities. But how are linear inequalities in two variables used in solving real-life problems? You will find these out in the activities in the next section. Before performing these activities, read and understand first important notes on linear inequalities in two variables and the examples presented. A linear inequality in two variables is an inequality that 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 both not 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 Certain situations in real life can be modeled by linear inequalities. Examples: 1. The total amount of 1-peso coins and 5-peso coins in the bag is more than Php 150. The situation can be modeled by the linear inequality x + 5y > 150, where x is the number of 1-peso coins and y is the number of 5-peso coins. 2. Emily bought two blouses and a pair of pants. The total amount she paid for the items is not more than Php 980. The situation can be modeled by the linear inequality 2x + y ≤ 980, where x is the cost of each blouse and y is the cost of a pair of pants. The graph of a linear inequality in two variables is the set of all points in the rectangular coordinate system whose ordered pairs satisfy the inequality. 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. 206 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 + 4 y=x+4 b. y < x – 2 y=x–2 c. y ≥ -x + 3 y = -x + 3 d. y ≤ -x – 5 y = -x – 5 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 > symbol, use a dashed or 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. a. y > x + 4 c. y ≥ -x + 3 b. y < x – 2 d. y ≤ -x – 5 207  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 the inequality 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 + 4 c. y ≥ -x + 3 For example, points (0, 3), (2, 2), and For example, points (-2, 8), (0, 7), and (4, -5) do not satisfy the inequality y > x + 4. (8, -1) satisfy the inequality y ≥ -x + 3. Therefore, the half-plane that does not Therefore, the half-plane containing contain these points will be shaded. these points will be shaded. The shaded portion constitutes the The shaded portion constitutes the solution of the linear inequality. solution of the linear inequality. b. y < x – 2 d. y ≤ -x – 5 Learn more about Linear Inequalities in Two Variables through the WEB. You may open the following links. 1. http://library.think- quest.org/20991/alg /systems.html 2. http://www.kgsepg. com/project-id/6565- inequalities-two-vari- able 3. http://www.monterey- institute.org/courses/ Algebra1/COURSE_ TEXT_RESOURCE/ U05_L2_T1_text_fi- nal.html 4. http://www.phschool. com/atschool/acade- my123/english/acad- emy123_content/wl- book-demo/ph-237s. html For example, points (0, 5), (-3, 7), and (2, 10) For example, points (12, -3), (0, -9), and (3, -11) 5. http://www.purple- do not satisfy the inequality y < x – 2. satisfy the inequality y ≤ -x – 5. math.com/modules/ ineqgrph.html Therefore, the half-plane that does not Therefore, the half-plane containing these 6. http://math.tutorvista. contain these points will be shaded. points will be shaded. com/algebra/linear- equations-in-two- The shaded portion constitutes the solution The shaded portion constitutes the solution of variables.html of the linear inequality. the linear inequality. 208  Now that you learned about linear inequalities in two variables and their graphs, you may now try the activities in the next section. What to What to Process Process Your goal in this section is to learn and understand key concepts of linear inequalities in two variables including their graphs and how they are used in real-life situations. Use the mathematical ideas and the examples presented in answering the activities provided. A ctivity 7 THAT’S ME! Directions: Tell which of the following is a linear inequality in two variables. Explain your answer. 1. 3x – y ≥ 12 6. -6x = 4 + 2y 2. 19 < y 7. x + 3y ≤ 7 2 3. y= x 8. x > -8 5 4. x ≤ 2y + 5 9. 9(x – 2) < 15 5. 7(x - 3) < 4y 10. 13x + 6 < 10 – 7y ES TIO a. How did you identify linear inequalities in two variables? How about those which are not linear inequalities in two variables? ? QU NS b. What makes a mathematical statement a linear inequality in two variables? c. Give at least 3 examples of linear inequalities in two variables. Describe each. How did you find the activity? Were you able to identify linear inequalities in two variables? In the next activity, you will determine if a given ordered pair is a solution of a linear inequality. 209 A ctivity 8 WHAT’S YOUR POINT? Directions: State whether each given ordered pair is a solution of the inequality. Justify your answer. 1. 2x – y > 10; (7, 2) 6. -3x + y < -12; (0, -5) 2. x + 3y ≤ 8; (4, -1) 7. 9 + x ≥ y; (-6, 3) 3. y < 4x – 5; (0, 0) 8. 2y – 2x ≤ 14; (-3, -3) 1 1 4. 7x – 2y ≥ 6; (-3, -8) 9. x + y > 5; (4, ) 2 2 2 1 5. 16 – y > x; (-1, 9) 10. 9x + y < 2; ( ,1) 3 5 ES TIO a. How did you determine if the given ordered pair is a solution of the ? QU NS inequality? b. What did you do to justify your answer? From the activity done, were you able to determine if the given ordered pair is a solution of the linear inequality? In the next activity, you will determine if the given coordinates of points on the graph satisfy an inequality. A ctivity 9 COME AND TEST ME! Directions: Tell which of the given coordinates of points on the graph satisfy the inequality. Justify your answer. 1. y < 2x + 2 a. (0, 2) b. (5, 1) c. (-4, 6) d. (8, -9) e. (-3, -12) 210  2. 3x ≥ 12 – 6y a. (1, -1) b. (4, 0) c. (6, 3) d. (0, 5) e. (-2, 8) 3. 3y ≥ 2x – 6 5. 2x + y > 3 a. (0, 0) b. (3, -4) c. (0, -2) d. (-9, -1) e. (-5, 6) 4. -4y < 2x - 12 a. (2, 4) b. (-4, 5) c. (-2, -2) d. (8.2, 5.5) 1 e. (4, ) 2 211  2x + y > 3 5. 1 a. (1 , 0) 2 b. (7, 1) c. (0, 0) d. (2, -12) e. (-10, -8) ES TIO a. How did you determine if the given coordinates of points on the ? QU NS graph satisfy the inequality? b. What did you do to justify your answer? Were you able to determine if the given coordinates of points on the graph satisfy the inequality? In the next activity, you will shade the part of the plane divider where the solutions of the inequality are found. A ctivity 10 COLOR ME! Directions: Shade the part of the plane divider where the solutions of the inequality is found. 1. y < x + 3 2. y–x>–5 212  3. x ≤ y – 4 5. 2x + y < 2 4. x +y≥1 ES TIO a. How did you determine the part of the plane to be shaded? ? QU NS b. Suppose a point is located on the plane where the graph of a linear inequality is drawn. How do you know if the coordinates of this point is a solution of the inequality? c. Give at least 5 solutions for each linear inequality. From the activity done, you were able to shade the part of the plane divider where the solutions of the inequality are found. In the next activity, you will draw and describe the graph of linear inequalities. 213 A ctivity 11 GRAPH AND TELL… Directions: Show the graph and describe the solutions of each of the following inequalities. Use the Cartesian coordinate plane below. 1. y > 4x 2. y > x + 2 3. 3x + y ≤ 5 1 4. y< x 3 5. x – y < -2 ES TIO a. How did you graph each of the linear inequalities? ? QU NS b. How do you describe the graphs of linear inequalities in two variables? c. Give at least 3 solutions for each linear inequality. d. How did you determine the solutions of the linear inequalities? Were you able to draw and describe the graph of linear inequalities? Were you able to give at least 3 solutions for each linear inequality? In the next activity, you will determine the linear inequality whose graph is described by the shaded region. 214 A ctivity 12 NAME THAT GRAPH! Directions: Write a linear inequality whose graph is described by the shaded region. 1. 4. 2. 5. 3. 215  ES TIO a. How did you determine the linear inequality given its graph? ? QU NS b. What mathematics concepts or principles did you apply to come up with the inequality? c. When will you use the symbol >, ,