6.1. Graphing Linear Inequalities in Two Variables. INVESTIGATE the Math. Reflecting

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1 6.1 Graphing Linear Inequalities in Two Variables YOU WILL NEED graphing technolog OR graph paper, ruler, and coloured pencils EXPLORE For which inequalities is (3, 1) a possible solution? How do ou know? a) b) 5 # c) 1, 1 d) $ 9 GOAL Solve problems b modelling linear inequalities in two variables. INVESTIGATE the Math Amir owns a health-food store. He is making a miture of nuts and raisins to sell in bulk. His supplier charges $5/kg for nuts and $8/kg for raisins.? What quantities of nuts and raisins can Amir mi together if he wants to spend less than $ to make the miture? A. Suppose that Amir wants to spend eactl $ to make the miture. Work with a partner to create an equation that represents this situation. B. To what set of numbers does the domain and range of the two variables in our equation belong? Use this information to help ou graph the equation on a coordinate plane. C. Eplain wh the graph is a line segment, not a ra or a line. D. What region of the coordinate plane includes points representing quantities of nuts and raisins that Amir could use if he wants to spend less than $? How do ou know? E. There are man possible solutions to Amir s problem. Plot at least three points that represent reasonable solutions to Amir s problem. Eplain wh ou chose these points. Reflecting solution set The set of all possible solutions. F. Discuss and then decide whether the solution set for Amir s problem is represented b i) points in the region above the line segment. ii) points in the region below the line segment. iii) points on the line segment. G. Wh might the line segment be considered a boundar of the solution set? H. Wh might ou use a dashed line segment for this graph instead of a solid line segment? 9 Chapter 6 Sstems of Linear Inequalities NEL

2 APPLY the Math eample 1 Solving a linear inequalit graphicall when it has a continuous solution set in two variables Graph the solution set for this linear inequalit: 1 5 $ 1 Robert s Solution: Using graph paper Linear equation that represents the boundar: I knew that the graph of the linear equation would form the boundar of the linear inequalit 1 5 $ 1. The variables represent numbers from the set of real numbers. [ R and [ R -intercept: () The -intercept is at (, ). -intercept: () The -intercept is at (5, ). The domain and range are not stated and no contet is given, so I assumed that the domain and range are the set of real numbers. This means that the solution set is continuous. I knew that I needed to plot and join onl two points to graph the linear equation. I decided to plot the two intercepts. To determine the -intercept, I substituted for. To determine the -intercept, I substituted for. continuous A connected set of numbers. In a continuous set, there is alwas another number between an two given numbers. Continuous variables represent things that can be measured, such as time. NEL 6.1 Graphing Linear Inequalities in Two Variables 95

3 solution region The part of the graph of a linear inequalit that represents the solution set; the solution region includes points on its boundar if the inequalit has the possibilit of equalit. 6 ( 5, ) (, ) 6 Since the linear inequalit has the possibilit of equalit ($), and the variables represent real numbers, I knew that the solution region includes all the points on its boundar. That s wh I drew a solid green line through the intercepts. -6 half plane The region on one side of the graph of a linear relation on a Cartesian plane. Test (, ) in 1 5 $ 1. LS RS () 1 5() 1 Since is not greater than or equal to 1, (, ) is not in the solution region. I needed to know which half plane, above or below the boundar, represents the solution region for the linear inequalit. To find out, I substituted the coordinates of a point in the half plane below the line. I used (, ) because it made the calculations simple. I alread knew that the solution region includes points on the boundar, so I didn t need to check a point on the line. Tip Communication If the solution set to a linear inequalit is continuous and the sign includes equalit (# or $), a solid green line is used for the boundar, and the solution region is shaded green, as shown to the right Since m test point below the boundar was not a solution, I shaded the half plane that did not include m test point. This was the region above the boundar. I used green shading to show that the solution set belongs to the set of real numbers. Since the domain and range are in the set of real numbers, I knew that the solution set is continuous. Therefore, the solution region includes all points in the shaded area and on the solid boundar. 96 Chapter 6 Sstems of Linear Inequalities NEL

4 Janet s Solution: Using graphing technolog 1 5 $ 1 5 $ $ 5 5 $ 5 1 The variables represent numbers from the set of real numbers. [ R and [ R To enter the linear inequalit into m graphing calculator, I had to isolate. I didn t have to reverse the inequalit sign because I didn t divide or multipl b a negative value. The domain and range are not stated and no contet is given, so I assumed the domain and range are in the set of real numbers. This means that the solution set is continuous. I entered the inequalit into the calculator and graphed it. I knew that the boundar was correctl shown because a solid line means equalit is possible. To verif whether the correct half plane was shaded, I used (1, ) in the half plane above the boundar as a test point. Test (1, ) in 1 5 $ 1. LS RS (1) 1 5() 1 18 Since 18 is greater than 1, (1, ) is in the solution region. Since (1, ) is a solution to the linear inequalit, I knew that the half plane above the boundar should be shaded. The solution region includes all the points in the shaded area and along the boundar because the solution set is continuous. Your Turn Compare the graphs of the following relations. What do ou notice? 1 5 $ , 1 NEL 6.1 Graphing Linear Inequalities in Two Variables 97

5 eample Graphing linear inequalities with vertical or horizontal boundaries Graph the solution set for each linear inequalit on a Cartesian plane. a) {(, )., [ R, [ R} b) {(, ) $ 6 1, [ I, [ I} Wnn s Solution a).. I isolated so I could graph the inequalit. The variables represent numbers from the set of real numbers. [ R and [ R The domain and range are stated as the set of real numbers. The solution set is continuous, so the solution region and its boundar will be green in m graph I drew the boundar of the linear inequalit as a dashed green line because I knew that the linear inequalit (.) does not include the possibilit of being equal to. Communication Tip If the solution set to a linear inequalit is continuous and the sign does not include equalit (, or.), a dashed green line is used for the boundar and the solution region is shaded green, as shown to the right I needed to decide which half plane to shade. For to be greater than, I knew that an point to the right of the boundar would work. The solution region includes all the points in the shaded area because the solution set is continuous. The solution region does not include points on the boundar. 98 Chapter 6 Sstem of Linear Inequalities NEL

6 b) $ 6 1 $ 1 # 1 # 3 Since the linear inequalit has onl one variable,, I isolated the. As I rearranged the linear inequalit, I divided both sides b. That s wh I reversed the sign from $ to #. The variables represent integers. [ I and [ I {(, ) $ 6 1, [ I, [ I} Your Turn The domain and range are stated as being in the set of integers. I knew this means that the solution set is discrete. I knew that points with integer coordinates below the line 5 3 were solutions, so I shaded the half plane below it orange. I knew the linear inequalit includes 3, so points on the boundar with integer coordinates are also solutions to the linear inequalit. I stippled the boundar and the orange half plane with green points to show that the solution set is discrete. The solution region includes onl the points with integer coordinates in the shaded region and along the boundar. How can ou tell if the boundar of a linear inequalit is vertical or horizontal without graphing the linear inequalit? Eplain. discrete Consisting of separate or distinct parts. Discrete variables represent things that can be counted, such as people in a room. Communication Tip If the solution set to a linear inequalit is discrete, the solution region is shaded orange and stippled with green points. If the sign includes equalit ($ or #), the boundar is also stippled. An eample of this is shown to the left. If equalit is not possible (, or.), the boundar is a dashed orange line. An eample of this is shown below NEL 6.1 Graphing Linear Inequalities in Two Variables 99

7 eample 3 Solving a real-world problem b graphing a linear inequalit with discrete whole-number solutions A sports store has a net revenue of $1 on ever pair of downhill skis sold and $1 on ever snowboard sold. The manager s goal is to have a net revenue of more than $6 a da from the sales of these two items. What combinations of ski and snowboard sales will meet or eceed this dail sales goal? Choose two combinations that make sense, and eplain our choices. Jerr s Solution The relationship between the number of pairs of skis,, the number of snowboards,, and the dail sales can be represented b the following linear inequalit: The variables represent whole numbers. [ W and [ W I defined the variables in this situation and wrote a linear inequalit to represent the problem. I knew that onl whole numbers are possible for and, since stores don t sell parts of skis or snowboards. Because the domain and range are restricted to the set of whole numbers, I knew that the solution set is discrete. I also knew that m graph would occur onl in the first quadrant. I isolated so I could enter the inequalit into m graphing calculator. I adjusted the calculator window to show onl the first quadrant, since the domain and range are both the set of whole numbers. The boundar is a dashed line, which means that the solution set does not include values on the line. {(, ) , [ W, [ W} 3 Chapter 6 Sstems of Linear Inequalities NEL

8 Test (, ) in LS RS 1() 1 1() 6 Since is not greater than 6, (, ) is not in the solution region. I used the test point (, ) to verif that the correct half plane was shaded. Since (, ) is not a solution to the linear inequalit, I knew that the half plane that did not include this point should be shaded. This was done correctl. When I interpreted the graph, I considered the contet of the problem. I knew that Test (, ) in LS RS 1() 1 1() Since 88. 6, (, ) is a solution. onl discrete points with whole-number coordinates in the solution region made sense. points along the dashed boundar are not part of the solution region. points with whole-number coordinates along the -ais and -ais boundaries are part of the solution region. I picked two points in the solution region, (, ) and (5, 3), as possible solutions to the problem. I verified that each point is a solution to the linear inequalit. Test (5, 3) in LS RS 1(5) 1 1(3) Since 86. 6, (5, 3) is a solution. Sales of four pairs of skis and four snowboards or sales of five pairs of skis and three snowboards will eceed the manager s net revenue goal of more than $6 a da. Some points in the solution region are more reasonable than others. For eample, the point (1, 1) is a valid solution, but it might be an unrealistic sales goal. Your Turn a) Would raising the dail sales goal to at least $1 change the graph that models this situation? Eplain. b) State two combinations of ski and snowboard sales that would meet or eceed this new dail sales goal. NEL 6.1 Graphing Linear Inequalities in Two Variables 31

9 In Summar Ke Idea When a linear inequalit in two variables is represented graphicall, its boundar divides the Cartesian plane into two half planes. One of these half planes represents the solution set of the linear inequalit, which ma or ma not include points on the boundar itself. Need to Know To graph a linear inequalit in two variables, follow these steps: Step 1. Graph the boundar of the solution region. If the linear inequalit includes the possibilit of equalit (# or $), and the solution set is continuous, draw a solid green line to show that all points on the boundar are included. If the linear inequalit includes the possibilit of equalit (# or $), and the solution set is discrete, stipple the boundar with green points. If the linear inequalit ecludes the possibilit of equalit (, or.), draw a dashed line to show that the points on the boundar are not included. - Use a dashed green line for continuous solution sets. - Use a dashed orange line for discrete solution sets. Step. Choose a test point that is on one side of the boundar. Substitute the coordinates of the test point into the linear inequalit. If possible, use the origin, (, ), to simplif our calculations. If the test point is a solution to the linear inequalit, shade the half plane that contains this point. Otherwise, shade the other half plane. - Use green shading for continuous solution sets. - Use orange shading with green stippling for discrete solution sets. For eample, {(, ) l # 1 5, [ R, [ R} {(, ) l. 1 5, [ R, [ R} {(, ) l $ 1 5, [ I, [ I} {(, ) l, 1 5, [ I, [ I} When interpreting the solution region for a linear inequalit, consider the restrictions on the domain and range of the variables. - If the solution set is continuous, all the points in the solution region are in the solution set. - If the solution set is discrete, onl specific points in the solution region are in the solution set. This is represented graphicall b stippling. - Some solution sets ma be restricted to specific quadrants. For eample, most linear inequalities representing real-world problem situations have graphs that are restricted to the first quadrant. 3 Chapter 6 Sstems of Linear Inequalities NEL

10 CHECK Your Understanding 1. Graph the solution set for each linear inequalit. a), 1 b), Consider the graph of this inequalit Make each of the following decisions, and provide our reasoning. a) whether the boundar should be dashed, stippled, or solid b) whether the half plane above or below the boundar should be shaded c) whether each point is in its solution region: i) (1, 1) ii) (1, ) iii) (1, ) 3. Bets and Flnn work at an ice cream stand. If Bets worked three times as man hours as she usuall does and Flnn worked twice the number of hours that he usuall does, together the would work less than 5 h. The situation can be modelled b the following linear inequalit: 3b 1 f, 5 a) What do the variables b and f represent? b) What restrictions does the contet place on the variables? Eplain. c) Suppose ou were to graph the inequalit. i) Describe the boundar. ii) Would ou shade the half plane above or below the boundar? iii) Would our graph involve all four quadrants? Eplain. d) What does a solution to this inequalit represent? PRACTISING. Match each graph with its linear inequalit, and justif our match. a) b) 6 c) i) {(, ) 3., [ W, [ W} ii) {(, ). 3, [ R, [ R} iii) {(, ) 3 $, [ R, [ R} 5. Graph the solution set for each linear inequalit. a). 1 8 d) 8. b) 3 # e) 1 1, c), 6 f ) 1 3 $ 1 NEL 6.1 Graphing Linear Inequalities in Two Variables 33

11 6. Graph the solution set for each linear inequalit. a) {(, ) $ , [ W, [ W} b) {(, ) 1 6 1,, [ I, [ I} c) {(, ) 5 #, [ W, [ W} d) {(, ) 1 # 5 1, [ I, [ I} e) {(, )., [ R, [ R} f ) {(, ) 5, 1, [ R, [ R} 7. Grace s favourite activities are going to the movies and skating with friends. She budgets herself no more than $75 a month for entertainment and transportation. Movie admission is $9 per movie, and skating costs $5 each time. A student bus pass for the month costs $5. a) Define the variables and write a linear inequalit to represent the situation. b) What are the restrictions on the variables? How do ou know? c) Graph the linear inequalit. Use our graph to determine: i) a combination of activities that Grace can afford and still have some mone left over ii) a combination of activities that she can afford with no mone left over iii) a combination of activities that will eceed her budget 8. Eamon coaches a hocke team of 18 plaers. He plans to bu new practice jerses and hocke sticks for the team. The supplier sells practice jerses for $5 each and hocke sticks for $85 each. Eamon can spend no more than $3 in total. He wants to know how man jerses and sticks he should bu. a) Write a linear inequalit to represent the situation. b) Use our inequalit to model the situation graphicall. c) Determine a reasonable solution to meet the needs of the team, and provide our reasoning. 9. For ever tedd bear that is sold at a fundraising banquet, $1 goes to charit. For ever ticket that is sold, $3 goes to charit. The organizers goal is to raise at least $5. The organizers need to know how man tedd bears and tickets must be sold to meet their goal. a) Define the variables and write a linear inequalit to represent the situation. b) What are the restrictions on the variables? How do ou know? c) Graph the linear inequalit to help ou determine whether each of the following points is in the solution set. The first coordinate is the number of tedd bears and the second is the number of tickets. i) (, ) ii) (5, 98) iii) (156, 15) 3 Chapter 6 Sstems of Linear Inequalities NEL

12 1. On Earth Da, a nurser sold more than $15 worth of maple and birch trees. The maple trees were sold for $75, and the birch trees were sold for $5. a) Define the variables and write a linear inequalit to represent the possible combinations of trees sold. Are there an restrictions on the variables? Eplain. b) Graph the linear inequalit. c) Use our graph to determine: i) if the nurser could have sold 13 of each tpe of tree ii) if 1 of one tpe and 9 of the other tpe could have been sold 11. In the fall, Javier plants tulip and crocus bulbs. Each tulip takes up an area of at least 1 square inches, and each crocus takes up an area of at least 9 square inches. Javier has a total area of 36 in. b 5 in., and he wants to plant at least 3 of each tpe of flower. He wants to know eactl how man of each tpe of flower he should plant. a) If ou were to graph Javier s situation, would the boundar be a dashed line, a stippled line, or a solid line? How do ou know? b) Graph the linear inequalit, and determine a reasonable solution to Javier s problem. 1. A banquet room is set up to seat, at most, 66 people. Each rectangular table seats 1 people, and each circular table seats 8 people. a) Define the variables and write a linear inequalit to represent the number of each tpe of table needed. Then graph our inequalit. b) The organizers of the banquet would like to have as close to the same number of rectangular tables and circular tables as possible. What combination of tables could the use? Eplain our choice. Closing 13. Joelle used deductive reasoning to conclude that the graph on the right represents a linear inequalit. a) What evidence has she used to arrive at this conclusion? b) State some other things ou know about this inequalit. Provide our reasoning for each Etending 1. Debbie and Gavin are moving. The have household goods that occup a volume of, at most, 16 cubic feet. Packing boes are available in two sizes: cubic feet and 6 cubic feet. The are sold in sets of four. a) Use a graph to determine what combinations of boes Debbie and Gavin could bu. b) If Debbie and Gavin wanted to use the least number of boes, what is the best combination? Eplain our thinking. 6 8 NEL 6.1 Graphing Linear Inequalities in Two Variables 35