Linear Programming. You can model sales with the following objective function. Sales 100x 50y. x 0 and y 0. x y 40

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1 Lesson 9.7 Objectives Solve systems of linear inequalities. Solving Systems of Inequalities Suppose a car dealer nets $500 for each family car (F) sold and $750 for each sports car (S) sold. The dealer must sell at least two family cars for every sports car and must earn at least $3,500 per week. You can find S and F by graphing the two inequalities that meet these conditions. Inequality F 2S 500F 750S 3,500 Meaning sell at least 2 family cars for each sports car make at least $3,500 per week if net for each family car sold is $500 and net for each sports car sold is $750 The solution of this system of inequalities is the intersection of the solution regions of the two inequalities. Family Cars Sold Activity F F = 2S (2, 4) F 750S 3,500 1 S Sports Cars Sold Intersecting Solution Sets see margin 1 Graph the lines F 2S and 500F 750S 3,500 on the same axes. 2 Use test points to determine the solution region for each inequality. 3 Find the area where the two solution regions intersect. This is the solution region for the system of inequalities. A solution region contains many solutions. In the car dealer problem, the dealer would want to know which solution maximizes profit. You can use linear programming to determine the optimum number of items to sell to maximize profits. Linear Programming Linear programming is a process used to find the optimum solution by graphing inequalities. An optimum solution is usually a minimum value or maximum value. An objective function is used to find the optimum solution. The linear inequalities, or constraints, limit the set of values of the variables. A feasible solution is a vertex of the shaded region when graphing constraints. 9.7 Solving Systems of Inequalities 531

2 Example Linear Programming A bicycle shop made $100 on each Model X bike sold and $50 on each Model Y bike sold. How can the owner s maximize sales? Solution You can model sales with the following objective function. Sales 100x 50y In the objective function, x is the number of Model X bikes sold and y is the number of Model Y sold. Suppose that each month the owner can assemble no more than 20 Model X bikes and no more than 30 Model Y bikes. These constraints are represented by inequalities. x 20 and y 30 Furthermore, suppose the maximum number of bikes the shop can assemble is 40. This constraint is written as an inequality. x y 40 Because the owner cannot assemble fewer than zero of either model, two final constraints also exist. x 0 and y 0 y (0, 30) (10, 30) y = (20, 20) x + y = y = 0 (0, 0) (20, 0) x Graph all 5 constraints. The shaded region is a polygon bounded by the five lines that represent the constraints. All feasible solutions to the problem are contained in the shaded region. Note particularly the five corner points around the perimeter of the polygon: (0, 0); (20, 0); (20, 20); (10, 30); and (0, 30). To find the optimum solution, substitute each of these five coordinate pairs into the objective function (sales equation) and compare the sales. x = 0 x = 20 Feasible Solution 0, 0 20, 0 20, 20 10, 30 0, 30 Objective Function (Sales) 100(0) 50(0) 0 100(20) 50(0) 2, (20) 50(20) 3, (10) 50(30) 2, (0) 50(30) 1,500 The table shows that the maximum monthly sales for the bike shop is $3,000. This occurs when the bike shop assembles and sells 20 of each model each month. Although any point in the shaded region of the graph of the system of inequalities satisfies all five inequalities (constraints), only the corner point (20, 20) maximizes the sales. 532 Chapter 9 Inequalities

3 Problem Solving Using the Four-Step Plan A pet store sells frogs for $4 and turtles for $6. The store must buy at least 30 frogs and at least 30 turtles from its supplier, but can buy no more than 90 total. How many of each pet should be bought to maximize the store s sales? (Assume every pet purchased will be sold.) see margin Step 1 Understand the Problem What is the store s goal? What are its constraints? Step 2 Develop a Plan Problem-solving strategy: Use a system of inequalities. How many unknowns are there in the problem? What are they? How many inequalities are needed to state all the problem constraints? Step 3 Carry Out the Plan Define variables. Write the inequalities. Graph the inequalities and shade the region that represents the solution. Use the shaded region and its boundaries to find the optimum solution. Step 4 Check the Results Make sure you have answered the question and that your solution satisfies the words of the problem. Substitute the solution into the inequalities and make sure they are true. If all the pets are sold, what is the total amount of sales? Lesson Assessment Think and Discuss 1. What conditions must be true for a system of inequalities to have a common solution? 2. If the lines for two inequalities are parallel, can there be a common solution? Why or why not? 9.7 Solving Systems of Inequalities 533

4 Practice and Problem Solving Graph each system of inequalities. see margin 3. 2x y 4 4. x y x y 4 y 2x 4. x y 2. y 2x x 3y x 5y x 7y 4 3x 2y 12. 3x 4y 1. 2x 3y x 2y y 6x y 3x 0 4y 3x 2. 3y 5x 6. 3y 2x A painting contractor estimates it will take 10 hours to paint a one-story house and 20 hours to paint a two-story house. The contractor submits a bid to paint 20 houses in less than 250 hours. a. Write a system of equations to model the time to paint the houses and the number of houses to be painted. x y 20; 10x 20y 250 b. Graph the system. see margin c. Give five whole-number solutions to the system. Explain what these solutions mean. (20, 0), (19, 1), (18, 2), (17, 3), (16, 4); see margin for explanation 13. It costs 50 cents to make a bracelet and $1 to make a necklace. To make a profit, the total cost for bracelets and necklaces must be less than $10. The jeweler can make no more than 14 pieces of jewelry each day. a. Write a system of inequalities to model the number of bracelets and necklaces to be made each day. b n 14; 0.5b n 10 b. Graph the system. see margin c. The jeweler sells bracelets for $3 and necklaces for $4. Test the three corner points on your graph and determine how many bracelets and necklaces should be made to maximize profits. 6 necklaces and 8 bracelets 534 Chapter 9 Inequalities

5 Write a system of inequalities for each situation. Graph the solution. 14. A farmer needs to enclose a rectangular field with wire fencing. He has 800 meters of fencing available. The length of the field must be greater than 100 meters. What are the possible dimensions of the field? What is the maximum possible area of the field? L W 400; L > 100; 40,000 square meters; see margin for graph 15. A manufacturer makes running shoes and basketball shoes. It costs $20 to make a pair of running shoes and $30 to make a pair of basketball shoes. The manufacturer can spend no more than $2,000 making both types of shoes each day. The company needs to make no fewer than 50 pairs of running shoes each day. How many of each type can be produced each day? 50 r 100; 0 b 33; see margin for graph 16. A hardware distributor wants to sell at least 45 electric and gas mowers each week. The profit on an electric mower is $50 and the profit on a gas mower is $40. Which combinations will give the distributor a profit of up to $2,500 per week? see margin Mixed Review 17. A box contains 10 gold medals, 15 silver medals, and 25 bronze medals. One medal is removed at random. What is the probability that a. the medal is silver? b. the medal is gold or bronze? 18. A box contains 10 gold medals, 15 silver medals, and 25 bronze medals. Two medals are removed at random. What is the probability that a. both medals are silver? b. one medal is gold and the other is bronze? 19. A box contains 10 gold medals, 15 silver medals, and 25 bronze medals. One medal is removed at random. It is replaced, and another medal is removed at random. What is the probability that a. both medals are the same? b. neither medal is bronze? Solve using a system of linear equations. 20. Sherrill has $8,000 invested in CD accounts. One account pays 6% simple interest, and the other pays 8% simple interest. This year Sherrill earned $580 interest on her two accounts. How much is invested in each account? $5,000 at 8%; $3,000 at 6% 9.7 Solving Systems of Inequalities 535