Problem 39 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The solution set of a system of linear inequalities in two variables is bounded if it can be enclosed by a rectangle.

Step-by-Step Solution

Answer

The given statement is true. The solution set of a system of linear inequalities in two variables is bounded if it can be enclosed by a rectangle because any solution set that can be enclosed by a rectangle will have at least two linear inequalities defining the horizontal boundaries and two defining the vertical boundaries, ensuring that the solution set is bounded within a finite region of the plane.

Step 1: Evaluating the given statement

Let's consider the given statement and try to interpret it in terms of linear inequalities: "The solution set of a system of linear inequalities in two variables is bounded if it can be enclosed by a rectangle." We can hypothesize that if the solution set can be enclosed by a rectangle, there are at least two linear inequalities defining the horizontal boundaries (top and bottom) and two defining the vertical boundaries (left and right).

Step 2: Testing the statement with an example

Let's test the given statement with an example of a system of linear inequalities that can be enclosed by a rectangle: Inequality 1: \( y \leq x + 2 \) (top boundary) Inequality 2: \( y \geq x - 2 \) (bottom boundary) Inequality 3: \( y \leq -x + 4 \) (right boundary) Inequality 4: \( y \geq -x \) (left boundary) It is easy to see that the solution set satisfies all four inequalities and forms a parallelogram, which can be enclosed by a rectangle. In this case, the statement appears to be true.

Step 3: Verifying the statement for a general case

However, we need to verify if the statement holds for any system of linear inequalities that can be enclosed by a rectangle. It can be proved that if there exists a rectangle enclosing the solution set for a system of linear inequalities, the solution set will always be bounded. Therefore, based on our analysis, the given statement is indeed true for any system of linear inequalities that can be enclosed by a rectangle. So, the statement is true.

Step 4: Conclusion

In conclusion, the statement "The solution set of a system of linear inequalities in two variables is bounded if it can be enclosed by a rectangle" is true. This is because any solution set that can be enclosed by a rectangle will have at least two linear inequalities defining the horizontal boundaries and two defining the vertical boundaries, which ensure that the solution set is bounded within a finite region of the plane.

Key Concepts

Bounded Solution Set

Understanding what a bounded solution set is can be vital for solving and graphing systems of linear inequalities. A bounded solution set occurs when all the solutions to a system of inequalities lie within a finite region of the coordinate plane.

Take the example of a rectangle on a graph that encompasses all possible solutions to a set of inequalities. Each side of the rectangle represents a boundary created by an inequality. If the system of inequalities is such that every solution fits within this rectangle, then we consider the solution set to be bounded. No matter how far you extend the sides of your rectangle, as long as it still contains all solutions, there is no point outside of it that satisfies the system. This is significant because it indicates that we're dealing with a limited set of possibilities rather than an infinite array.

Applying the Concept

When approaching homework problems, look for signs of boundaries that can enclose the solution set. This can simplify the process of determining the nature of the solution set and aid in visualizing the solutions.

Linear Algebra

At its core, linear algebra is about understanding and solving equations that describe lines, planes, and their interactions—in this case, through systems of linear inequalities. These inequalities are just like equations but with a '<' or '>' sign instead of an '='. They reflect a range of possible solutions, rather than a single point.

Diving into Systems of Inequalities

In linear algebra, working with systems of inequalities means finding where these inequalities overlap. A system might consist of several linear inequalities, and the collective solutions to a system are what we're interested in finding. This process often involves a good deal of visualization, which is why graphing plays a major role and why your understanding of a bounded solution set is crucial. Remember, the solutions to these systems are not just individual points, but whole regions of the graph where all the inequalities hold true.

Graphical Representation of Inequalities

The graphical representation of inequalities is a tool that helps us visualize the solutions to a system of linear inequalities. By plotting each inequality on the graph, we can visually determine where the solutions to the system lie.

Starting with a single inequality, we shade the area that represents all the possible solutions. With a system of inequalities, we look for the intersection of these shaded areas. This intersection is the set of all points that satisfy all inequalities simultaneously.

Intersecting Regions

For example, when we graph the inequalities from our exercise, we find overlapping shaded regions. The boundaries may take the form of lines, and the solutions lie in the area where the lines intersect. As more inequalities are added to the system, the solution set can become more restricted until it is fully 'bounded' by the intersecting regions (potentially resembling a shape like a rectangle).

For students, visualizing these regions can significantly improve their understanding of how inequalities form a solution set, and practice with these visualizations is a great way to become more comfortable with linear algebra concepts.