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Notes 7.5 – Systems of Inequalities

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I. Half-Planes A.) Given the inequality, the line is the boundary, and the half- plane “below” the boundary is the solution set.

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B.) Ex.1 – Solve the following system (1, -1) (-2, -4)

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C.) Ex. 2– Solve the following system of inequalities.

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The vertices for the solution to the system are and the solution set is area bounded by the lines through these points.

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II. Linear Programming A.) Def. - Given an objective function f n where In two dimensions, the function takes on the form f = ax + by. The solution is called the feasible region and the constraints are a system of inequalities.

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B.) Ex. 3–Find the maximum and minimum values of the objective function f = 2x + 9y subject to the following constraints

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Minimum ValueMaximum Value

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Johnson’s Produce is purchasing fertilizer with two nutrients: N (nitrogen) and P (phosphorous). They need at least 180 units of N and 90 units of P. Their supplier has two brands of fertilizer for them to buy. Brand A costs $10 a bag and has 4 units of N and 1 unit of P. Brand B costs $5 a bag and has 1 unit of each nutrient. Johnson’s Produce can pay at most $800 for the fertilizer. How many bags of each brand should be purchased to minimize their cost? C.) Ex. 4– Example 7 from the text on page 619

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Let x = # of bags of A and y = # of bags of B Therefore, our objective function for the COST is with constraints of

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Solution set looks like the following graph

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Minimum Value The vertices of the region are

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