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Chapter 3: Problem 21

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded. $$ \begin{array}{r} x-y \leq 0 \\ 2 x+3 y \geq 10 \end{array} $$

Short Answer

Expert verified
The solution set for the given system of inequalities is the intersection of shaded regions above the lines \(y=x\) and \(y=-\frac{2}{3}x + \frac{10}{3}\). The feasible region is open, meaning the solution set is unbounded.

Step by step solution

01

Rewrite the inequalities

Rewrite the inequalities in slope-intercept form (y = mx + b) to make them easier to graph. For the first inequality, \(x - y \leq 0\) \(y \geq x\) For the second inequality, \(2x + 3y \geq 10\) \(3y \geq -2x + 10\) \(y \geq -\frac{2}{3}x + \frac{10}{3}\)
02

Graph the inequalities

Now that we have the inequalities in the slope-intercept form, we can graph them. First, graph the lines that are the boundary of each inequality: 1. Boundary line for the first inequality is \(y = x\). 2. Boundary line for the second inequality is \(y = -\frac{2}{3}x + \frac{10}{3}\).
03

Identify the feasible region

To determine the feasible region, we will use shading. For each inequality, we will shade the area that satisfies the inequality: 1. First inequality: Shade the region above the line \(y = x\). 2. Second inequality: Shade the region above the line \(y = -\frac{2}{3}x + \frac{10}{3}\). The intersection of the shaded regions is the feasible region. This is the region where both inequalities are true simultaneously.
04

Determine bounded or unbounded

Now, to determine whether the solution set is bounded or unbounded, observe the resulting feasible region. If the region is closed (no open ends), the solution set is bounded. If the region is open (has at least one open end), the solution set is unbounded. In this case, the feasible region is open, and thus, the solution set is unbounded.

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Most popular questions from this chapter

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The problem $$ \begin{aligned} \text { Minimize } & C=2 x+3 y \\ \text { subject to } & 2 x+3 y \leq 6 \\ & x-y=0 \\ & x \geq 0, y \geq 0 \end{aligned} $$ is a linear programming problem.

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Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. An optimal solution of a linear programming problem is a feasible solution, but a feasible solution of a linear programming problem need not be an optimal solution.

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