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Chapter 3: Problem 28

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded. $$ \begin{aligned} x+y & \geq 20 \\ x+2 y & \geq 40 \\ x \geq 0, y & \geq 0 \end{aligned} $$

Short Answer

Expert verified
The solution set for the given system of inequalities is the quadrilateral enclosed by the points (0,20), (20,0), (40,0), and (0,40) in the first quadrant. The solution set is bounded since it is contained within a finite region.

Step by step solution

01

Graph each inequality

First, we need to graph each inequality on the coordinate plane. To make it easier to graph, we'll rewrite the inequalities as equalities and find the x and y-intercepts for each equation. 1. \(x+y=20\): x-intercept: set y=0, then x=20 y-intercept: set x=0, then y=20 2. \(x+2y=40\): x-intercept: set y=0, then x=40 y-intercept: set x=0, then y=20 3. \(x\geq 0\), \(y\geq0\): This represents the first quadrant, where both x and y are non-negative. Now, plot the lines corresponding to equalities and shade the regions where the inequalities are satisfied.
02

Identify the solution region

The solution region is the area where all inequalities are satisfied simultaneously. 1. For \(x+y \geq 20\), we have the region above and including the line \(x+y=20\). 2. For \(x+2y\geq 40\), we have the region above and including the line \(x+2y=40\). 3. For \(x \geq 0\) and \(y \geq 0\), we have the first quadrant. The solution region is where these three regions overlap, which is the quadrilateral enclosed by the points (0,20), (20,0), (40,0), and (0,40).
03

Bounded or Unbounded Solution Set

The solution set is bounded if it is contained within a finite region or unbounded if it extends infinitely in any direction. Since the solution region is a quadrilateral enclosed by the points (0,20), (20,0), (40,0), and (0,40), it is a finite region. Therefore, the solution set is bounded.

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Most popular questions from this chapter

You are given a linear programming problem. a. Use the method of corners to solve the problem. b. Find the range of values that the coefficient of \(x\) can assume without changing the optimal solution. c. Find the range of values that resource 1 (requirement 1) can assume. d. Find the shadow price for resource 1 (requirement 1). e. Identify the binding and nonbinding constraints. $$ \begin{array}{cc} \text { Minimize } & C=3 x+4 y \\ \text { subject to } & x+3 y \geq 8 \\ & x+y \geq 4 \\ & x \geq 0, y \geq 0 \end{array} $$

Solve each linear programming problem by the method of corners. $$ \begin{array}{rr} \text { Minimize } & C=3 x+4 y \\ \text { subject to } & x+y \geq 3 \\ & x+2 y \geq 4 \\ & x \geq 0, y \geq 0 \end{array} $$

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded. $$ \begin{aligned} x-y & \geq-6 \\ x-2 y & \leq-2 \\ x+2 y & \geq 6 \\ x-2 y & \geq-14 \\ x \geq 0, y & \geq 0 \end{aligned} $$

Solve each linear programming problem by the method of corners. $$ \begin{array}{ll} \text { Maximize } & P=4 x+2 y \\ \text { subject to } & x+y \leq 8 \\ & 2 x+y \leq 10 \\ & x \geq 0, y \geq 0 \end{array} $$

Solve each linear programming problem by the method of corners. $$ \begin{aligned} \text { Maximize } & P=2 x+5 y \\ \text { subject to } & 2 x+y \leq 16 \\ & 2 x+3 y \leq 24 \\ y & \leq 6 \\ & x \geq 0, y & \geq 0 \end{aligned} $$
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