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Then/Now You solved systems of linear inequalities by graphing. Find the maximum and minimum values of a function over a region. Solve real-world optimization problems using linear programming.

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Vocabulary linear programming feasible region bounded unbounded optimize

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Concept

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Example 1 Bounded Region 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 f(x, y) = 3x – 2y for this region. x ≤ 5 y ≤ 4 x + y ≥ 2 Step 1 Graph the inequalities. The polygon formed is a triangle with vertices at (–2, 4), (5, –3), and (5, 4).

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Example 1 Bounded Region 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). ← minimum ← maximum

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Example 1 A.maximum: f(4, 5) = 5 minimum: f(1, 5) = –11 B.maximum: f(4, 2) = 10 minimum: f(1, 5) = –11 C.maximum: f(4, 2) = 10 minimum: f(4, 5) = 5 D.maximum: f(1, 5) = –11 minimum: f(4, 2) = 10 Graph the following system of inequalities. What are the maximum and minimum values of the function f(x, y) = 4x – 3y for the feasible region of the graph? x ≤ 4y ≤ 5 x + y ≥ 6

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Concept

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Example 3 Optimization with Linear Programming 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.

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Example 3 Optimization with Linear Programming Step 1Define the variables. m = the number of mowing jobs p = the number of pruning jobs

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Example 3 Optimization with Linear Programming Step 2Write 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. 1m + 3p ≤ 9 There are no more than 2 pruning jobs a day. p ≤ 2

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Example 3 Optimization with Linear Programming Step 3Graph the system of inequalities.

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Example 3 Optimization with Linear Programming 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 f(m, p) = 40m + 120p. We want to find the maximum value for this function.

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Example 3 Optimization with Linear Programming Step 6Substitute the coordinates of the vertices into the function. Step 7Select the greatest amount.

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Example 3 Optimization with Linear Programming 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 mowings and 2 prunings or 9 mowings and 0 prunings.

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