Problem 21 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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

Step-by-Step Solution

Answer

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 1: 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, \(2x + 3y \geq 10\) \(3y \geq -2x + 10\) \(y \geq -\frac{2}{3}x + \frac{10}{3}\)

Step 2: 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 \(y = -\frac{2}{3}x + \frac{10}{3}\).

Step 3: 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.

Step 4: 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.

Key Concepts

Graphical Solution

Solving a system of inequalities graphically involves plotting lines on a coordinate plane that represent each inequality in the system. We start by rewriting the inequalities in slope-intercept form:

• For example, the inequality \(x - y \leq 0\) becomes \(y \geq x\).
• Similarly, \(2x + 3y \geq 10\) becomes \(y \geq -\frac{2}{3}x + \frac{10}{3}\).

Having the inequalities in slope-intercept form simplifies the process because this form is easy to graph. The slope-intercept form \(y = mx + b\) shows you the slope \(m\) of the line and the y-intercept \(b\) where the line crosses the y-axis.
For each inequality, we draw a boundary line on a graph. If an inequality is "greater than or equal to" or "less than or equal to," the boundary line is solid, indicating that points on the line satisfy the inequality.
After graphing each line, you determine the feasible region by shading the side of the line where the inequality holds true. The overlapping shaded region represents the set of solutions to the system of inequalities.

Bounded and Unbounded Regions

Once you graph the inequalities, observe the shaded area that represents possible solutions. This region is known as the feasible region. Depending on the boundaries created by the inequalities, the feasible region can be

• Bounded: Enclosed on all sides by lines or curves, forming a closed shape such as a triangle or polygon. Such regions contain all solutions within a finite area.
• Unbounded: Open on at least one side, extending infinitely in that direction. An unbounded region means the solutions extend infinitely and the feasible region does not form a closed shape.

In our example, we noticed the shaded area from the inequalities extends infinitely upwards and to the right, meaning there are no boundaries enclosing the feasible region completely. Therefore, the solution set is unbounded, indicating the set of solutions is infinite in extent.

Slope-Intercept Form

The slope-intercept form, \(y = mx + b\), is an equation format where a straight line is described by two main components: the slope \(m\) and the y-intercept \(b\). Understanding this form is crucial for graphing and solving inequalities. Here’s what each component represents:

• The slope \(m\) indicates how steep the line is. It tells us how much \(y\) changes for a unit change in \(x\). If \(m\) is positive, the line rises as \(x\) increases; if negative, the line falls as \(x\) increases.
• The y-intercept \(b\) shows where the line crosses the y-axis. This starting point helps in sketching the initial portion of the line on the graph.

To tackle inequalities, convert each inequality into the slope-intercept form. This enables you to easily plot the line and decide where the shaded regions belong. For example, in our exercise, converting \(2x + 3y \geq 10\) into \(y \geq -\frac{2}{3}x + \frac{10}{3}\) enables a straightforward graphing process, showcasing both the slope and the y-intercept for clarity.