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Chapter 5 Linear Inequalities and Linear Programming

Chapter 5 Linear Inequalities and Linear Programming. Section 2 Systems of Linear Inequalities in Two Variables. Solving Systems of Linear Inequalities Graphically. We now consider systems of linear inequalities such as x + y > 6 2 x – y > 0

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Chapter 5 Linear Inequalities and Linear Programming

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  1. Chapter 5Linear Inequalities and Linear Programming Section 2 Systems of Linear Inequalities in Two Variables

  2. Solving Systems of Linear Inequalities Graphically • We now consider systems of linear inequalities such as x + y > 6 2x – y > 0 • We wish to solve such systems graphically, that is, to find the graph of all ordered pairs of real numbers (x, y) that simultaneously satisfy all the inequalities in the system. • The graph is called the solution region for the system (or feasible region.) • To find the solution region, we graph each inequality in the system and then take the intersection of all the graphs.

  3. Graphing a System of Linear Inequalities: Example To graph a system of linear inequalities such as we proceed as follows: Graph each inequality on the same axes. The solution is the set of points whose coordinates satisfy all the inequalities of the system. In other words, the solution is the intersection of the regions determined by each separate inequality.

  4. The graph of the first inequality y < –(1/2)x + 2 consists of the region shaded yellow. It lies below the dotted liney = –(1/2)x + 2. The graph of the second inequality is the blue shaded region is above the solid linex – 4 = y. The graph is the region which is colored both blue and yellow. Graph of Example

  5. Corner Points A corner point of a solution region is a point in the solution region that is the intersection of two boundary lines. In the previous example, the solution region had a corner point of (4,0) because that was the intersection of the lines y = –1/2 x + 2 and y = x – 4. Corner point

  6. Bounded and Unbounded Solution Regions A solution region of a system of linear inequalities is bounded if it can be enclosed within a circle. If it cannot be enclosed within a circle, it is unbounded. The previous example had an unbounded solution region because it extended infinitely far to the left (and up and down.) We will now see an example of a bounded solution region.

  7. Graph of More Than Two Linear Inequalities To graph more than two linear inequalities, the same procedure is used. Graph each inequality separately. The graph of a system of linear inequalities is the area that is common to all graphs, or the intersection of the graphs of the individual inequalities. Example:

  8. Application Suppose a manufacturer makes two types of skis: a trick ski and a slalom ski. Suppose each trick ski requires 8 hours of design work and 4 hours of finishing. Each slalom ski requires 8 hours of design and 12 hours of finishing. Furthermore, the total number of hours allocated for design work is 160, and the total available hours for finishing work is 180 hours. Finally, the number of trick skis produced must be less than or equal to 15. How many trick skis and how many slalom skis can be made under these conditions? How many possible answers? Construct a set of linear inequalities that can be used for this problem.

  9. Let x represent the number of trick skis and y represent the number of slalom skis. Then the following system of linear inequalities describes our problem mathematically. Actually, only whole numbers for x and y should be used, but we will assume, for the moment that x and y can be any positive real number. ApplicationSolution x and y must both be positive Number of trick skis has to be less than or equal to 15 Constraint on the total number of design hours Constraint on the number of finishing hours

  10. ApplicationGraph of Solution The origin satisfies all the inequalities, so for each of the lines we use the side that includes the origin. The intersection of all graphs is the yellow shaded region. The solution region is bounded and the corner points are (0,15), (7.5, 12.5), (15, 5), and (15, 0)

  11. Example 4: Application In one week, Ed can mow at most 9 times and rake at most 7 times. He charges $20 for mowing and $10 for raking. He needs to make more than $125 in one week. Show and describe all the possible combinations of mowing and raking that Ed can do to meet his goal. List two possible combinations. Earnings per Job ($) Mowing 20 Raking 10

  12. Example 4 Continued Step 1 Write a system of inequalities. Let x represent the number of mowing jobs and y represent the number of raking jobs. x ≤ 9 He can do at most 9 mowing jobs. y ≤ 7 He can do at most 7 raking jobs. 20x + 10y > 125 He wants to earn more than $125.

  13. Solutions Example 4 Continued Step 2 Graph the system. The graph should be in only the first quadrant because the number of jobs cannot be negative.

  14. Example 4 Continued Step 3 Describe all possible combinations. All possible combinations represented by ordered pairs of whole numbers in the solution region will meet Ed’s requirement of mowing, raking, and earning more than $125 in one week. Answers must be whole numbers because he cannot work a portion of a job. Step 4 List the two possible combinations. Two possible combinations are: 7 mowing and 4 raking jobs 8 mowing and 1 raking jobs

  15. Helpful Hint An ordered pair solution of the system need not have whole numbers, but answers to many application problems may be restricted to whole numbers.

  16. Check It Out! Example 4 At her party, Alice is serving pepper jack cheese and cheddar cheese. She wants to have at least 2 pounds of each. Alice wants to spend at most $20 on cheese. Show and describe all possible combinations of the two cheeses Alice could buy. List two possible combinations. Price per Pound ($) Pepper Jack 4 Cheddar 2

  17. Check It Out! Example 4 Continued Step 1 Write a system of inequalities. Let x represent the pounds of cheddar and y represent the pounds of pepper jack. x ≥ 2 She wants at least 2 pounds of cheddar. y ≥ 2 She wants at least 2 pounds of pepper jack. 2x + 4y ≤ 20 She wants to spend no more than $20.

  18. Solutions Check It Out! Example 4 Continued Step 2 Graph the system. The graph should be in only the first quadrant because the amount of cheese cannot be negative.

  19. Step 3 Describe all possible combinations. All possible combinations within the gray region will meet Alice’s requirement of at most $20 for cheese and no less than 2 pounds of either type of cheese. Answers need not be whole numbers as she can buy fractions of a pound of cheese. Step 4 Two possible combinations are (2, 3) and (4, 2.5). 2 cheddar, 3 pepper jack or 4 cheddar, 2.5 pepper jack

  20. Lesson Quiz: Part I y < x + 2 1. Graph . 5x + 2y ≥ 10 Give two ordered pairs that are solutions and two that are not solutions. Possible answer: solutions: (4, 4), (8, 6); not solutions: (0, 0), (–2, 3)

  21. Lesson Quiz: Part II 2. Dee has at most $150 to spend on restocking dolls and trains at her toy store. Dolls cost $7.50 and trains cost $5.00. Dee needs no more than 10 trains and she needs at least 8 dolls. Show and describe all possible combinations of dolls and trains that Dee can buy. List two possible combinations.

  22. Solutions Lesson Quiz: Part II Continued Reasonable answers must be whole numbers. Possible answer: (12 dolls, 6 trains) and (16 dolls, 4 trains)

  23. Notes Over 3.3 7. An arena contains 1200 seats. For an upcoming concert, tickets will be priced $12.00 for some seats and $10.00 for others. At least 500 tickets are to be priced at $10.00, and the total sales must be at least $7200 to make a profit. What are the possible ways to price the tickets. Writing and Using a System of Inequalities Write and graph a system of linear inequalities to describe the problem.

  24. Notes Over 3.3 Graphing a System of Three Inequalities Graph the system of linear inequalities. Solid lines False True False

  25. Problem Model Patricio’s family, on average, drives their SUV more than twice as many miles as they drive their car. His family’s car emits 0.75 pounds of CO2 per mile and the SUV emits 1.25 pounds of CO2 per mile. Patricio is concern with the environment and convinces his family to limit the total CO2 emissions to less than 600 pounds per month. How many miles can they drive their car and SUV to meet this limit? x = SUV miles x > 2y = Car miles y 0.75y + 1.25x < 600

  26. x > 2y 0.75y + 1.25y < 600 Problem Model

  27. Problem Model The science club can spend at most $400 on a field trip to a dinosaur exhibit. It has enough chaperones to allow at most 100 students to go on the trip. The exhibit costs $3.00 for students 12 and under and $6.00 for students 12 and over. How many students 12 years and under can go if 20 students over 12 go? x = Students 12 and under x + y ≤ 100 = Students 12 and over y 3x + 4y ≤ 400

  28. Problem Model x + y ≤ 100 3x + 4y ≤ 400

  29. Now you try… The Math Club want to advertise their fundraiser each week in the school paper. They know that a front-page ad is more effective than an ad inside the paper. They have a total of $30 budget for advertising. It costs $2 for each front-page ad and $1 for each inside-page ad. If the club wants to advertise at least 20 times, what are the different possibilities for the number of front-page and inside-page ads. x = front-page ads x + y ≤ 20 = inside-page ads y 2x + 1y ≤ 30

  30. Now you try… x + y ≤ 20 2x + 1y ≤ 30

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