Problem 23 - 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. $$ \begin{aligned} x+2 y & \geq 3 \\ 2 x+4 y & \leq-2 \end{aligned} $$

Step-by-Step Solution

Answer

The solution set for the system of inequalities is the overlapping shaded region, represented by a triangle formed by the graph of the inequalities: $$ \begin{aligned} y & \geq -\frac{1}{2}x + \frac{3}{2} \\\ y & \leq -\frac{1}{2}x - \frac{1}{2} \end{aligned} $$ The solution set is bounded, as the area of this triangle is finite.

Step 1: Convert Inequalities to Slope-Intercept Form

We'll begin by converting both inequalities into slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. For the first inequality, \(x + 2y \geq 3\), we solve for \(y\): 1. Subtract \(x\) from both sides: \(2y \geq -x + 3\). 2. Divide both sides by 2: \(y \geq -\frac{1}{2}x + \frac{3}{2}\). For the second inequality, \(2x + 4y \leq -2\), we also solve for \(y\): 1. Divide both sides by 2: \(x + 2y \leq -1\). 2. Subtract \(x\) from both sides: \(2y \leq -x -1\). 3. Divide both sides by 2: \(y \leq -\frac{1}{2}x - \frac{1}{2}\). So we have the following inequalities in slope-intercept form: $$ \begin{aligned} y & \geq -\frac{1}{2}x + \frac{3}{2} \\\ y & \leq -\frac{1}{2}x - \frac{1}{2} \end{aligned} $$

Step 2: Graph the Inequalities

Now that we have the inequalities in slope-intercept form, we can graph them as two lines. The first inequality, \(y \geq -\frac{1}{2}x + \frac{3}{2}\), represents a line with slope -1/2 and y-intercept 3/2. Since the inequality is greater than or equal to, we will illustrate this line with a solid line and shade the region above the line. For the second inequality, \(y \leq -\frac{1}{2}x - \frac{1}{2}\), we have a line with the same slope as the previous one, -1/2, but with a y-intercept of -1/2. The inequality symbol is less than or equal to, meaning we will be illustrating this line with a solid line and shade the area below the line. Our solution set is the area where both shaded regions overlap.

Step 3: Determine if the Solution Set is Bounded or Unbounded

Looking at the graph, we can see that the overlapping shaded region is in the shape of a triangle, with vertices at points where the lines intersect. This means that the solution set is bounded because the area of this triangle is finite. In conclusion, the graphically represented solution set for the given system of inequalities is bounded, and the solution set is the region where the shading from both inequalities overlaps.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a common way to write linear equations and is crucial for graphing inequalities. It is expressed as:

• \( y = mx + b \)

Here, \( m \) represents the slope of the line, indicating its steepness and direction, while \( b \) is the y-intercept, showing where the line crosses the y-axis. When solving inequalities graphically, converting them to this form allows us to easily draw the lines on a coordinate plane.
In the given system, through rearranging and simplifying both inequalities, we get:

• \( y \geq -\frac{1}{2}x + \frac{3}{2} \)
• \( y \leq -\frac{1}{2}x - \frac{1}{2} \)

Both expressions are now in slope-intercept form, making them easier to graph.

Boundary Lines

Boundary lines are the lines which represent the limits of our inequalities on the graph. When graphing an inequality, the boundary line is drawn to show the edge of the region that satisfies the inequality. In our examples, both boundary lines are derived from the converted slope-intercept form equations.
Boundary lines can be solid or dashed.

• A solid line indicates that points on the line are included in the solution set (inequality symbols \( \leq \) or \( \geq \)).
• A dashed line means the line itself is not included (inequality symbols \( < \) or \( > \)).

In this case, both boundary lines are solid because the inequalities are \( \geq \) and \( \leq \). These lines act as closed borders of the colored regions on a graph, defining the area where potential solutions can exist.

Solution Set

The solution set is the region on the graph where the solutions to the inequalities lie. This is determined by observing where the shaded regions of the inequalities overlap. The overlapping region is where **both** inequalities' conditions are satisfied simultaneously.
For our specific inequalities:

• Shading above the line \( y \geq -\frac{1}{2}x + \frac{3}{2} \)
• Shading below the line \( y \leq -\frac{1}{2}x - \frac{1}{2} \)
These give us a triangular region where both conditions meet, representing all the (x, y) pairs that satisfy the system.
Since the overlapping region forms a triangle, the solution set is finite and said to be **bounded**, meaning it does not extend infinitely in any direction.

Inequality Systems

Inequality systems involve dealing with two or more inequalities at once. Our task is to find where these inequalities are true at the same time. In other words, we look for solutions that make both inequality conditions hold.
To solve inequality systems graphically:

• Convert each inequality to slope-intercept form for easy plotting.
• Draw each equation as a boundary line on the coordinate plane.
• Shade the appropriate regions for each inequality.
• Identify the area where the shaded regions overlap. This is the solution set.

Handling systems like these often reveals important information, such as whether a solution set is bounded or unbounded, and the shape it forms on a graph. Each step toward shading and interpretation leads us to a clearer view of how inequalities interact on a coordinate plane.