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Chapter 3: Problem 26

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded. $$ \begin{array}{l} 4 x-3 y \leq 12 \\ 5 x+2 y \leq 10 \\ x \geq 0, y \geq 0 \end{array} $$

Short Answer

Expert verified
The solution set for the system of inequalities is the region where all shaded regions overlap, including the region where \(x \geq 0\) and \(y \geq 0\). Inspecting the graph, we notice that the overlapped region has boundaries on all sides, so the solution set is bounded.

Step by step solution

01

Rewrite the inequalities in slope-intercept form

To rewrite the inequalities in slope-intercept form, we need to solve for y. Original inequalities: $$ \begin{array}{l} 4 x-3 y \leq 12 \\\ 5 x+2 y \leq 10 \\\ x \geq 0, y \geq 0 \end{array} $$ Solving for y in each inequality: $$ \begin{array}{l} -3y \leq -4x + 12 \\ 2y \leq -5x + 10 \\ x \geq 0, y \geq 0 \end{array} $$ Now, divide by the coefficients of y: $$ \begin{array}{l} y \geq \frac{4}{3}x-4 \\ y \leq -\frac{5}{2}x+5 \\ x \geq 0, y \geq 0 \end{array} $$
02

Plot the lines on a graph

In order to create the graph and plot the lines, follow these steps: 1. Sketch a coordinate plane and label it. 2. Draw the line y = (4/3)x - 4, and shade the region above the line. 3. Draw the line y = (-5/2)x + 5, and shade the region below the line. 4. Mark the region where x 鈮� 0 and y 鈮� 0.
03

Determine the region where all inequalities are satisfied

Now, observe the graph and identify the region where all shaded regions overlap, including the region where x 鈮� 0 and y 鈮� 0. This overlapped region is the solution set of the system of inequalities.
04

Identify if the solution set is bounded or unbounded

Look at the overlapped region and check if it is surrounded by boundaries on all sides or if it extends infinitely in any direction. If it has boundaries on all sides, the solution set is bounded. Otherwise, it is unbounded.

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Most popular questions from this chapter

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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