Foundations of Mathematics 11

Transcription

1 6.1 Graphing Linear Inequalities in Two Variables (Part 1) Review of Graphing Linear Equations: Most linear equations will be written in one of two different forms: 0 General form Slope y-intercept form (slope = m & y-intercept = b) The method used to graph a linear relation without technology depends on the form in which the linear equation is written. Example 1: Use slope and y-intercept method to sketch: a) 2 3 b) 2 Example 2: Use x- and y-intercepts slope method to sketch: a) b) Equations with only one variable produce either a horizontal or vertical line. 105

2 Example 2. For which inequalities is (3, 1) a possible solution? a) 13 3 > 4 b) 2 5 c) < 10 d) 9 Try. Which point(s) is/are in the solution region of the inequality 3 5 8? a) (0, 0) b) (5, 1) c) (6, 2) d) (2,4) The following linear inequalities can be graphed on a Cartesian Plane: < 10 2 > 0 The solution region to a linear inequality in one or two variables can be represented on a coordinate plane using a boundary line and shading one side of the line. The boundary line will be solid or broken according to the following rules: A solid boundary line is used to represent or. A broken or dotted boundary line is used to represent > or <. The following procedure can be used to graph the solution region of a two variable linear inequality. Step 1: On a coordinate plane, graph the corresponding linear equation using a table of values, intercepts, or point and slope. Draw the line solid or broken according to the rule above. Step 2: The line divides the coordinate plane into two regions, called half planes. The solution region will be on one side of the line. To determine which side, choose the coordinates of a point not on the line, called a test point, and determine if the coordinates of the point satisfy the inequality. If the inequality is satisfied, then the solution is the region from which the point was chosen. If not, the solution region is the other region. Step 3: Shade the appropriate region. 106

3 Graph Linear Inequalities with Continuous Variables Example 3: Graph the solution set for the following linear inequalities a) (,) 2 > 6, #, #$ Slope and y-intercept Method x- and y-intercepts Method b) (,) , #, #$ 107

4 To graph a linear inequality with 0 or 0, the inequatlity must be re- plane. written in the form or. Example 4. Graph the solution set for each linear inequality on a Cartesian a) (,) 3 6 0, #, #$ b) (,) 4 2 > 0, #, #$ c) (,) 4 8 0, #, #$ d) (,) 6 3 < 0, #, #$ 108

5 Equations with only one variable produce either a horizontal or vertical line. Example 5. Graph the solution set for each linear inequality on a Cartesian plane. a) (,) 3 < 0, #, #$ b) (,) 2 0, #, #$ c) (,) 4 < 0, #, #$ d) (,) 5 0, #, #$ 109

6 Foundations of Mathematics Graphing Linear Inequalities in Two Variables (Part 2) Graph Linear Inequalities with Discrete Number Variables Example 1. Graph the solution set for the following inequalities. > 6, %, %$ 5 10, a) (, ) 2 b) (, ) 2 c) (, ) 2 > 7, ', d) (, ) 2 '$ , ', %, '$ %$

7 When interpreting the solution region for a linear inequality, consider the restrictions on the domain and range of the variables. 1. If the solution set is continuous, all the points in the solution region are in the solution set. 2. If the solution set is discrete, only specific points in the solution region are in the solution set. This is represented graphically by stippling. Graph Linear Inequalities Vertical or Horizontal Boundaries Example 2. Graph the solution set for each linear inequality on a Cartesian plane. a) (,) 2 > 0, %, %$ b) (,) 3 6 6, ', '$ c) (,) 7 14, %, %$ d) (,) 5 10, ', '$ 111

8 Try. Match each linear inequality to its graph. Most linear inequalities representing real-world problem situations have graphs that are restricted to the first quadrant because the values of the variables in the system must be positive. Solve By Graphing a Linear Inequality with Discrete Whole-Number Solutions Example 3. A sports store has net revenue of $100 on every pair of downhill skis sold and $120 on every snowboard sold. The manager s goal is to have net revenue of more than $600 a day from the sales of these two items. What combinations of ski and snowboard sales will meet or exceed this daily sales goal? Choose two combinations that make sense, and explain your choices. 112

9 6.2 Exploring Graphs of Systems of Linear Inequalities System of Linear Inequalities A set of two or more linear inequalities that are graphed on the same coordinate plane; the intersection of their solution regions represents the solution set for the system. Solve System of Linear Inequalities Example 1. Graph the system of linear inequalities. Justify your representation of the solution set. a) (,) 2 5, #, #$ and (,) 5, #, #$ b) (,), #, #$ and (,) 5, #, #$ 113

10 c) (,) < 10, ', '$ and (,) 2, ', '$ d) (,) 3 6, ', '$ and (,) 1, ', '$ 114

11 6.3 Graphing to Solve Systems of Linear Inequalities Solve Graphically a System of Linear Inequalities with Continuous Variables Example 1. Graph the solution set for the following system of inequalities, and choose two possible solutions from the set. a) 3 2 > 6 and 3 Use an open dot to show that an intersection point of a system s boundaries is excluded from the solution set. An intersection point is excluded when a dashed line intersects either a dashed or solid line. b) 2 1 > 3 and

12 Solve Graphically a System of Linear Inequalities with Discrete Whole-Number Variables Example 2. Graph the system and determine a possible solution. a) (,) 3 2 > 6, ', '$ and (,) 3, ', '$ b) (,) 10, (, ($ and (,) > 6, (, ($ Most linear inequalities representing real-world problem situations have graphs that are restricted to the first quadrant because the values of the variables in the system must be positive. 116

13 Example 3. A restaurant owner, Moon, has two part-time employees: K and J. K is skilled at cooking but has limited experience with customers. Moon pays her $18 an hour. J has experience with customers but not much with cooking. Moon pays him $10 an hour. Moon has a budget of $470 for their wages. Moon can hire both of these employees for no more than 30 hrs. a week, in total. Both employees are scheduled in whole numbers of hours. Determine two possible combinations of numbers of hours scheduled for K and J. Try. A company makes two types of very fancy cakes: cheesecakes and tiramisus. The company can make a maximum of 10 cakes in a day. Cheesecakes are more profitable than tiramisus; the company makes at least 3 more cheesecakes than tiramisus each day. Determine two possible combinations of numbers of cheesecakes and tiramisus can be made in one day. 117

14 6.4 Optimization Problems Part 1: Creating the Model Create a Model for an Optimization with Whole-Number Variables Example 1. On a flight between Winnipeg and Vancouver, there are business class and economy seats. - At capacity, the airplane can hold no more than 145 passengers. - No fewer than 130 economy seats (e) are sold, and no more than 8 business class seats (b) are sold. - The airline charges $615 for business class seats and $245 for economy seats. The flight company wants to know the combination of economy and business class seats that will generate the minimum and maximum revenues. Create a model to represent this situation and graph the feasible region. Objective Function 118

15 Example 2. A school is organizing a track and field meet. - There will be no more than 250 events and no fewer than 100 events to be scheduled. - The organizers allow 15 min. for each track event (t) and 45 min. for each field event (f). - They are considering different combinations of track and field events. The school wants to determine the least and greatest amounts of time they should allow. Create a model to represent this situation and graph the feasible region. Objective Function 119

16 Try. Sophie has two summer jobs. - She works no more than a total of 32 hrs. a week. Both jobs allow her to have flexible hours but in whole hours only. - At one job, Sophie works no less than 12 hrs. and earns $8.75/hr. At the other job, Sophie works no more than 24 hrs. and earns $9.00/hr. She would like to know the combination of numbers of hours that will allow her to maximize her earnings. Create a model to represent this situation and graph the feasible region. Objective Function 120

17 6.5 Optimization Problems Part 2: Exploring Solutions The solution to an optimization problem (maximum or minimum) is usually found at one of the vertices of the feasible region. Explore the Feasible Region of a System of Linear Inequalities Example 1. Ribbon flowers and crepe-paper rosettes are being made as decorations. - At least 50 ribbon flowers and no more than 75 rosettes are needed. - Altogether, no more than 140 decorations are needed - Each ribbon flower takes 6 min. to make, and each rosette takes 9 min. to make. What combinations of ribbon flowers and rosettes will take the least and the most amount of time to make? What are the minimum and maximum time needed to make these decorations? Objective Function 121

18 Example 2. A transportation company leases vehicles. - It has 10-passenger vans and 16-passenger minibuses to lease. - At most, 5 minibuses are available to lease. - There are 120 or fewer people to be transported. - Each minibus plus a driver costs $730 to lease, and each van plus a driver costs $550. What combinations of vans and minibuses will allow the transportation company to minimize and maximize the value of leases? What will the minimum and maximum values be? How many people can be transported? Objective Function 122

19 Try. The stylists in a hair salon cut hair for women and men. - The salon books at least 4 women s appointments for every man s appointment. - Usually there are 90 or more appointments, in total, during a week. - The salon is trying to reduce the number of hours the stylists work. - A women s cut takes about 75 min., and a man s cut takes about 30 min. What combination of women s and men s appointments would minimize and maximize the number of hours the stylists work? How many hours would this be? Objective Function 123

20 6.6 Optimization Problems Part 3: Linear Programming Solve Optimization Problems Example 1. A toy company manufactures two types of toy vehicles: racing cars and SUVs. - Because the supply of materials is limited, no more than 60 SUVs (s) and 40 racing cars (r) and can be made each day. - However, the company can make 70 or more vehicles, in total, each day. - It costs $12 to make a SUV (s) and $8 to make a racing car (r). There are many possible combinations of SUVs and racing cars that could be made. The company wants to know what combination will result in the minimum cost, and what cost that will be? Objective Function 124

21 Linear Programming A mathematical technique used to determine which solutions in the feasible region result in the optimal solutions of the objective function. Try. Chubby Cubbies Education Technologies (CCET) manufactures packages of pattern blocks and linking cubes. - CCET can produce at least 60 packages of pattern block and linking cubes per day. - Due to the amount of material at hand, CCET can produce at most 30 packages of pattern block and 50 packages of linking cubes per day. - The sale price of the pattern blocks is $7 per pack; the sale price of the linking cubes is $5 per pack. The company wants to know what combinations will result in the maximum revenue, and what revenue that would be. Objective Function 125

22 Example 2. L&G Construction is competing for a contract to build a fence. - The fence will be no longer than 50 yd. and will consist of narrow boards that are 6.in. wide and wide boards that are 8 in. wide. - There must be no fewer than 100 wide boards and no more than 80 narrow boards. - The narrow boards cost $3.56 each, and the wide boards costs $4.36 each. Determine the minimum cost for the lumber to build the fence. Objective Function 126