Math 141 Week-in-Review # 4 (Graphing Systems of Linear Inequalities and Solving Linear Programming Problems)

Brief Overview of Section 3.1:

• Procedure for Graphing Linear Inequalities:

– Draw the graph of the equation obtained for the given inequality by replacing the inequality sign with an equal sign.Use a dashed line if the problem involves a strict inequality, < or >. Otherwise, use a solid line to indicate that theline itself constitutes part of the solution.

– Pick a test point, (a,b), lying in one of the half-planes determined by the line sketched in Step 1 and substitute thenumbers a and b for the values of x and y in the given inequality. For simplicity, use the origin, (0,0), wheneverpossible.

– If the inequality is satisfied (True), the graph of the solution to the inequality is the half-plane containing the test point(Shade the region containing the test point). Otherwise (if the inequality is False), the solution is the half-plane notcontaining the test point (Shade the region that does not contain the test point).

1. Determine graphically the solution set for the following systems of inequalities. Indicate whether the solution set is boundedor unbounded.

(a) x+ y > 6y 10

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Math 141 WIR, c�Maya Johnson, Fall 2019

(b) 20x+10y � 10010x+20y � 10010x+10y � 80

x � 0, y � 0

Brief Overview of Section 3.3:

• Theorem 1: Solutions of Linear Programming Problems

– If a linear programming problem has a solution, then it must occur at a corner point of the feasible set, S,associated with the problem.

– If the objective function, P, is optimized at two adjacent corner points of S, then it is optimized at every point onthe line segment joining the two points (infinitely many solutions).

• Theorem 2: Existence of a Solution

Suppose we are given a linear programming problem with a feasible set S and an objective funtion P = ax+by.

– If S is bounded then P has both a maximum and a minimum value on S.– If S is unbounded and both a and b are nonnegative, then P has a minimum value on S provided that the constraints

defining S include the inequalities x � 0 and y � 0.– If S is empty, then the linear programming problem has no solution; that is, P has neither a maximum nor a

minimum value. We say that the problem is infeasible.

• The Method of Corners

– Graph the feasible set.– If the feasible set is nonempty, find the coordinates of all corner points of the feasible set. In this class we will

use the “rref” calculator function to find corner points whenever the points are where two lines are crossing.– Evaluate the objective function at each corner point.– Find the corner point(s) that renders the objective function a maximum (or minimum).

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Math 141 WIR, c�Maya Johnson, Fall 2019

2. Maximize: P = 3x+2ySubject to: 0.2x+0.1y 1

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x � 0, y � 0

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Math 141 WIR, c�Maya Johnson, Fall 2019

3. Maximize: P = 2x+3ySubject to: 5x+4y � 56

x+2y � 22x � 0, y � 0

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Math 141 WIR, c�Maya Johnson, Fall 2019

4. Minimize: C = 2x+4ySubject to: 0.1x+0.1y � 1

x+2y � 145x+7y 70x � 0, y � 0

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Math 141 WIR, c�Maya Johnson, Fall 2019

5. A serving of fruit salad has 3 grams of walnuts, 3 grams of fiber, and sells for $4. A serving of vegetable salad has 6 gramsof walnuts, 2 grams of fiber, and sells for $2.

(a) How many servings of each type of salad would maximize revenue if you have 24 grams of walnuts and 12 grams offiber available?

(b) Are there any leftover resources? Be specific.

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