Problem 28 - 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. $$ \begin{aligned} x+y & \geq 20 \\ x+2 y & \geq 40 \\ x \geq 0, y & \geq 0 \end{aligned} $$

Step-by-Step Solution

Answer

The solution set for the given system of inequalities is the quadrilateral enclosed by the points (0,20), (20,0), (40,0), and (0,40) in the first quadrant. The solution set is bounded since it is contained within a finite region.

Step 1: Graph each inequality

First, we need to graph each inequality on the coordinate plane. To make it easier to graph, we'll rewrite the inequalities as equalities and find the x and y-intercepts for each equation. 1. \(x+y=20\): x-intercept: set y=0, then x=20 y-intercept: set x=0, then y=20 2. \(x+2y=40\): x-intercept: set y=0, then x=40 y-intercept: set x=0, then y=20 3. \(x\geq 0\), \(y\geq0\): This represents the first quadrant, where both x and y are non-negative. Now, plot the lines corresponding to equalities and shade the regions where the inequalities are satisfied.

Step 2: Identify the solution region

The solution region is the area where all inequalities are satisfied simultaneously. 1. For \(x+y \geq 20\), we have the region above and including the line \(x+y=20\). 2. For \(x+2y\geq 40\), we have the region above and including the line \(x+2y=40\). 3. For \(x \geq 0\) and \(y \geq 0\), we have the first quadrant. The solution region is where these three regions overlap, which is the quadrilateral enclosed by the points (0,20), (20,0), (40,0), and (0,40).

Step 3: Bounded or Unbounded Solution Set

The solution set is bounded if it is contained within a finite region or unbounded if it extends infinitely in any direction. Since the solution region is a quadrilateral enclosed by the points (0,20), (20,0), (40,0), and (0,40), it is a finite region. Therefore, the solution set is bounded.

Key Concepts

Systems of Inequalities

When tackling systems of inequalities, we're dealing with multiple inequalities that we need to satisfy at the same time. Imagine a system like a set of rules in a game where each inequality is a rule that has to be followed. To find a solution, you must find values that make all the inequalities true simultaneously.

For the system given in the exercise, we are working with two linear inequalities and two more describing the positive sides of a coordinate plane. Think of it as you are trying to find a common meeting area that adheres to all the rules laid out by these 'boundary lines'. Graphically, the solution to the system of inequalities will be a region on the coordinate plane, and algebraically, it will be a set containing all the pairs of numbers (x, y) that satisfy all the inequalities together.

Coordinate Plane Graphing

Coordinate plane graphing is the heart of visualizing the relationships between numbers in algebra. The coordinate plane has two axes: the horizontal x-axis and the vertical y-axis, which divide the plane into four quadrants. Points on the plane are identified by their coordinates (x, y), which tell us their location relative to the two axes.

When graphing inequalities, like in our exercise, each inequality is graphed as if it were an equation, using the intercepts to find the line. Then we decide which side of the line to shade by considering the inequality sign. This is where your pencils (or digital tools) get busy with colors or patterns to illustrate which side of the line is included in the solution set. If an inequality is 'greater than', we shade above the line, and if it's 'less than', we shade below. For 'greater than or equal to' and 'less than or equal to', we also include the line itself in the shading.

Bounded Solution Set

A bounded solution set can be thought of like a fenced-off area; it has clear boundaries and doesn't go on forever. In contrast, an unbounded set is like an open field extending out to infinity. To determine whether a solution set is bounded, we look at where the shaded regions from all the inequalities overlap on the graph. If this overlapped area forms a closed shape (like a polygon), then the solution is bounded because it's confined to a finite space.

In our exercise, the solution set is the quadrilateral formed where all the shaded regions meet. Since this is a shape with clear edges and doesn’t extend to infinity, we identify the solution set as bounded. This is vital for many real-world applications as it represents scenarios with definite limits or constraints. Think of something like budgeting or designing a space where you have to work within certain financial or physical boundaries.