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Warm - Up

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Learning Targets I can solve systems of inequalities by graphing. I can determine the coordinates of the vertices of a region formed by the graph of a system of inequalities.

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Quick Review of Inequalities GREATER THAN GREATER THAN OR EQUAL TO DOTTED LINE SOLID LINE Remember when to shade above or below the line?

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Example 3-1a Solve the system of inequalities by graphing. solution of Regions 1 and 2 solution of Regions 2 and 3

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Example 3-1b Solve each system of inequalities by graphing. a. Answer:

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Example 3-2a Solve the system of inequalities by graphing. The graphs do not overlap, so the solutions have no points in common. Graph both inequalities. Answer: The solution set is .

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Example 3-2b Solve the system of inequalities by graphing. Answer:

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Example 3-4a Graph each inequality. The intersection of the graphs forms a triangle. Answer: The vertices of the triangle are at (0, 1), (4, 0), and (1, 3). Find the coordinates of the vertices of the figure formed byand

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Example 3-4b Answer: (–1, 1), (0, 3), and (5, –2)

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Big Idea: Explain what it means to find the vertices of a system of inequalities.

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Assignment: Practice

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Linear Programming

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Warm - Up Graph and find the vertices of the following system of inequalities. y > -4y < 2x + 22x + y < 6

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Homework Solutions

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Learning Targets I can determine the coordinates of the vertices of a region formed by the graph of a system of inequalities. I can find the maximum and minimum values of a function over a region.

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Vocabulary Constraints: Another term for “inequalities.” Feasible Region: The intersection of the graphs. Bounded: The polygonal region formed by the graphing of a system of constraints.

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Example 4-1a Step 1Find the vertices of the region. Graph the inequalities. The polygon formed is a triangle with vertices at (–2, 4), (5, –3), and (5, 4). Graph the following system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the function for this region.

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Example 4-1a Step 2 Use a table to find the maximum and minimum values of f(x, y). Substitute the coordinates of the vertices into the function. Answer:The vertices of the feasible region are (–2, 4), (5, –3), and (5, 4). The maximum value is 21 at (5, –3). The minimum value is –14 at (–2, 4). (x, y)3x – 2yf(x, y) (-2, 4) (5, -3) (5, 4) Plug each of your vertices into the equation 3(-2) – 2(4) -14 3(5) – 2(-3) 21 3(5) – 2(4) 7

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Example 4-1b Graph the following system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the function for this region. Answer: vertices: (1, 5), (4, 5), (4, 2)

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(x, y)4x – 3yf(x, y) (1, 5) (4, 5) (4, 2) Step 2 Use a table to find the maximum and minimum values of f(x, y). Substitute the coordinates of the vertices into the function. maximum: f(4, 2) = 10, minimum: f(1, 5) = –11 Plug each of your vertices into the equation 4(1) – 3(5)-11 4(4) – 3(5) 1 4(4) – 3(2) 10

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Vertices: (1, 0), (1, 4), (3, 0) (x, y)3x + yf(x, y) (1, 0) (1, 4) (3, 0) Max:Min:

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Example 4-2b Answer: vertices: (0, –3), (6, 0) ; maximum: f(6, 0) = 6 ; no minimum Graph the following system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the function for this region.

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Before your assignment… Remember the steps involved: 1. Find the vertices of the region. Graph the inequalities. 2. Use a table to find the maximum and minimum values of f(x, y). Substitute the coordinates of the vertices into the function.

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Assignment Practice Worksheet

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Linear Programming

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Homework Solutions

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Learning Target: I can solve real-world problems using linear programming.

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Steps to solving linear programming problems.

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Example 4-3a Landscaping A landscaping company has crews who mow lawns and prune shrubbery. The company schedules 1 hour for mowing jobs and 3 hours for pruning jobs. Each crew is scheduled for no more than 2 pruning jobs per day. Each crew’s schedule is set up for a maximum of 9 hours per day. On the average, the charge for mowing a lawn is $40 and the charge for pruning shrubbery is $120. Find a combination of mowing lawns and pruning shrubs that will maximize the income the company receives per day from one of its crews. Step 1 Define the variables. m = the number of mowing jobs p = the number of pruning jobs

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Example 4-3a Step 2 Write a system of inequalities. Since the number of jobs cannot be negative, m and p must be nonnegative numbers. m 0, p 0 Mowing jobs take 1 hour. Pruning jobs take 3 hours. There are 9 hours to do the jobs. There are no more than 2 pruning jobs a day. p 2

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Example 4-3a Step 3 Graph the system of inequalities.

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Example 4-3a Step 4Find the coordinates of the vertices of the feasible region. From the graph, the vertices are at (0, 2), (3, 2), (9, 0), and (0, 0). Step 5Write the function to be maximized. The function that describes the income is We want to find the maximum value for this function.

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Example 4-3a Step 6Substitute the coordinates of the vertices into the function. Step 7 Select the greatest amount. (m, p)(m, p) 40m + 120p f(m, p)f(m, p) (0, 2) 40(0) + 120(2)240 (3, 2) 40(3) + 120(2)360 (9, 0) 40(9) + 120(0)360 (0, 0) 40(0) + 120(0)0

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Example 4-3a Answer: The maximum values are 360 at (3, 2) and 360 at (9, 0). This means that the company receives the most money with 3 mows and 2 prunings or 9 mows and 0 prunings.

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Step 1: Define the variables: v = # of office visits s = # of surgeries Step 2: Write a system of inequalities Appointments cannot be negative, so v > 0 and s > 0 An office visit is 20 minutes, and surgery is 40 minutes. There are 7 hours available for appointments. 20v + 40s < 420 (7 hours = 420 minutes) The veterinarian cannot do more than 6 surgeries. s < 6

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Continued… Step 3: Graph the system of inequalities Step 4: Find the vertices. Step 5: Write a function to be maximized or minimized.

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Continued… Step 6: Substitute the coordinates of the vertices into the function. Step 7: Select the greatest amount. Answer: When Dolores schedules 6 surgeries and 9 office visits, she earns the highest amount per day at $1245.00 (s,v)125s + 55vf(s,v) (0,0) (6,0) (6,9) (0,21) 125(0) + 55(0)0 125(6) + 55(0)750 125(6) + 55(9)1245 125(0) + 55(21)1155 Maximum

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Example 4-3b Landscaping A landscaping company has crews who rake leaves and mulch. The company schedules 2 hours for mulching jobs and 4 hours for raking jobs. Each crew is scheduled for no more than 2 raking jobs per day. Each crew’s schedule is set up for a maximum of 8 hours per day. On the average, the charge for raking a lawn is $50 and the charge for mulching is $30. Find a combination of raking leaves and mulching that will maximize the income the company receives per day from one of its crews.

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List out the steps… 1. Define the variables 2. Write a system of inequalities.

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3. Graph the system of inequalities.

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4. Find the coordinates of the vertices of the feasible region. 5. Write a function to be maximized or minimized.

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6. Substitute the coordinates of the vertices into the function.

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7. Select the greatest or least result. Answer the problem.

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Example 4-3b Answer: 0 raking jobs and 4 mulching jobs

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Assignment 3.4B Linear Programming Worksheet

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Linear Programming Practice # 1 1.A test consists of short answer and essay problems. Short answer problems are worth six points, and essay problems are worth 10 points. (c) Write the max points equation : f(x, y) = It takes 2 minutes to answer the short answer problems and 4 minutes to answer the essay problems. You have 40 minutes for the test and you may choose no more than 12 problems to answer. Assuming you answer all of the problems you attempted correctly, how many of each type of problem should you answer to get the highest score?

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You should answer ____short answer and __ essay problems.

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2.Jimmy is baking cookies for a bake sale. He is making chocolate chip cookies and oatmeal raisin cookies. He gets $0.25 for each chocolate chip cookie and $0.30 for each oatmeal raisin cookie. (c) Write the profit equation : f(x, y) = He cannot make more than 500 cookies of each kind, and he cannot make more than 800 cookies total. How many of each kind of cookie should he make to get the most money?

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Jimmy should make ch. chip and ___oatmeal-raisin cookies.

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3) Jayne assembles two types of cameras: the Excellent and the Premier. Her cost of assembling these cameras is $21 for the Excellent and $42 for the Premier. The total funds available for her to use are $1,260. It takes 5 hours to assemble an Excellent and 2 hours to assemble a Premier. She can work for no more than 100 hours total. If the profit on the two cameras is $8 for the Excellent and $10 for the Premier, determine how many cameras of each kind should Jayne assemble to maximize profit?

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You should answer ____excellent and __ premier.

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4) A studio sells photographs and prints. It cost $20 to purchase each photograph and it takes 2 hours to frame it. It costs $25 to purchase each print and it takes 5 hours to frame it. The store has at most $400 to spend and at most 60 hours to frame. It makes $30 profit on each photograph and $50 profit on each print. Find the number of each that the studio should purchase to maximize profits.

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You should answer ____photographs and __ prints.

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5. A tent-maker makes a Standard model and an Expedition model. Each standard tent requires one hour of labor from the cutting department and 3 hours of labor from the assembly department. Each Expedition tent requires 2 hours of labor from the cutting department and 4 hours of labor from the assembly department. The maximum labor hours available per week in the cutting department is 32 and in the assembly department 84. If the company makes $50 profit on each Standard tent and $80 profit on each Expedition tent, how many tents of each type should be made to maximize weekly profit?

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You should answer ____Standard and __ Expedition. 21 16 2832 ?

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6.A carpenter makes bookcases in two sizes, large and small. It takes 6 hours to make a large bookcase and 2 hours to make a small one. The carpenter can spend only 24 hours per week making bookcases and must make at least two of each size per week. The profit on a large bookcase is $50 and the profit on a small bookcase is $20. How many of each size must be made per week in order to provide maximum profit?

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You should answer ____large and __ small. 12 4 2 2

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