Problem 29 - 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. $$ \begin{aligned} 3 x-7 y & \geq-24 \\ x+3 y & \geq 8 \\ x \geq 0, y & \geq 0 \end{aligned} $$

Step-by-Step Solution

Answer

The solution set is the area on the graph where all shaded regions intersect, bounded by the lines \(3x-7y=-24\), \(x+3y=8\), \(x=0\), and \(y=0\). This area forms a quadrilateral shape with a finite area, indicating the solution set is bounded.

Step 1: Convert inequalities into equations for plotting

Firstly, we will convert each inequality into an equation by replacing the inequalities signs with equal signs, like so: 1. \(3x - 7y = -24\) 2. \(x + 3y = 8\) 3. \(x = 0\) 4. \(y = 0\) Now we have four equations that will help us in drawing the boundary lines on the graph.

Step 2: Graph the boundary lines

We will now graph the lines representing the boundary of each inequality. 1. \(3x - 7y = -24\) → When x = 0: \(y = \frac{24}{7}\), When y = 0: \(x = -8\). Plot these points and draw the line passing through them. 2. \(x + 3y = 8\) → When x = 0: \(y = \frac{8}{3}\), When y = 0: \(x = 8\). Plot these points and draw the line passing through them. 3. \(x = 0\), which is the y-axis. 4. \(y = 0\), which is the x-axis.

Step 3: Determine the shading direction for each inequality

Next, we will select a random test point for each inequality (excluding the test points that lie on the boundary lines) and see if the inequality holds true or false. If it holds true, we will shade the region containing the test point; if it is false, we will shade the opposite region. 1. For \(3x - 7y \geq -24\): test point (0,0) \(3(0) - 7(0) \geq -24\) → \(0 \geq -24\), which is true. Shade the region containing the origin in this case. 2. For \(x + 3y \geq 8\): test point (0,0) \(0 + 3(0) \geq 8\) → \(0 \geq 8\), which is false. Shade the opposite region (not containing the origin). 3. For \(x \geq 0\): test point (1,0) \(1 \geq 0\), which is true. Shade the region containing (1,0) (to the right of y-axis). 4. For \(y \geq 0\): test point (0,1) \(1 \geq 0\), which is true. Shade the region containing (0,1) (above x-axis).

Step 4: Determine the solution set

Now, identify the area on the graph where all the shaded regions intersect. This area contains the solutions to all four inequalities. Mark this area on the graph.

Step 5: Determine if the solution set is bounded or unbounded

Observe the shape of the solution set. If the shape has a finite area, it is bounded. If it does not have a finite area, it is unbounded. In this case, the solution set has a finite area, formed by the intersection of the shaded regions. So, it is a bounded solution set.

Key Concepts

Systems of Inequalities

When dealing with systems of inequalities, you're working with multiple inequalities at the same time, rather than just a single equation or inequality. Solving these systems involves finding values that satisfy all inequalities in the system simultaneously. In this particular exercise, we have four inequalities:

• \(3x - 7y \geq -24\)
• \(x + 3y \geq 8\)
• \(x \geq 0\)
• \(y \geq 0\)

Each of these inequalities constrains the x and y values, defining a specific region on the graph where all conditions are met. The solution consists of all the points inside the overlapping region that satisfies every single inequality. Understanding each constraint's impact helps to effectively graph the system.

Boundary Lines

Boundary lines are an essential component when solving systems of inequalities graphically. Each inequality is turned into an equation when we replace inequality signs with equality signs. This transformation helps us to draw the associated line on the graph, which acts as the boundary for potential solutions. The boundary lines for this exercise are:

• \(3x - 7y = -24\)
• \(x + 3y = 8\)
• \(x = 0\)
• \(y = 0\)

To graph these lines, calculate where each line intersects the axes. For instance, the line \(3x - 7y = -24\) crosses the y-axis at \(y = \frac{24}{7}\) when \(x=0\) and the x-axis at \(x=-8\) when \(y=0\). Each boundary line forms a potential divider for shaded regions on the graph.

Shading Regions

After drawing the boundary lines, the next step is to determine which side of each line contains the solutions for the inequalities. This is done by shading regions on the graph. For each inequality, a test point not on the line, such as the origin (0,0), is selected to check which side satisfies the inequality.If the test point satisfies the inequality, you shade the region containing that point; if not, shade the opposite side. For example:

• For \(3x - 7y \geq -24\), since the origin satisfies the inequality, we shade the side including the origin.
• For \(x + 3y \geq 8\), the origin does not satisfy, so we shade the opposite side.
• For \(x \geq 0\), the region to the right of the y-axis is shaded.
• For \(y \geq 0\), the region above the x-axis is shaded.

The solution is found in the overlapping shaded area on the graph where all inequalities are satisfied simultaneously.

Bounded and Unbounded Sets

The final step in graphically solving systems of inequalities involves identifying whether the solution set is bounded or unbounded. A solution set is considered bounded if it forms a closed and finite area on the graph, encapsulated by the boundary lines. In this exercise, the intersecting shaded region is enclosed by the lines drawn, which means it has a finite area and is therefore a bounded set. A bounded set indicates that there are limits to the solutions, and the values cannot extend to infinity in any direction. On the other hand, an unbounded set lacks such constraints, allowing the shaded region to extend infinitely in one or more directions. Understanding whether a solution set is bounded helps you grasp the limitations (or lack thereof) on the possible solutions within the system.