Problem 20 - 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. $$ \begin{array}{r} 3 x-2 y>-13 \\ -x+2 y>5 \end{array} $$

Step-by-Step Solution

Answer

The solution set is the intersection of the shaded regions, which represents the values of x and y that satisfy both inequalities. In this case, the intersection is an unbounded region extending infinitely to the right. Therefore, the solution set is unbounded.

Step 1: Rewrite inequalities as equations.

First, let's rewrite the given inequalities as equations: 1. \(3x - 2y = -13\) 2. \(-x + 2y = 5\)

Step 2: Graph the equations.

Next, graph the equations on a coordinate plane. Equation 1: To graph \(3x - 2y = -13\), find the x and y-intercepts: - When x = 0: \(-2y=-13 \implies y=\dfrac{13}{-2}\) - When y = 0: \(3x=-13 \implies x=-\dfrac{13}{3}\) Plot these points (0,-6.5) and (-4.33,0) and draw a line through them. Equation 2: To graph \(-x + 2y = 5\), find the x and y-intercepts: - When x = 0: \(2y=5 \implies y=\dfrac{5}{2}\) - When y = 0: \(-x=5 \implies x=-5\) Plot these points (0,2.5) and (-5,0) and draw a line through them.

Step 3: Shade the Regions.

Now, shade the regions that satisfy each inequality. For inequality \(3x - 2y > -13\), test a point not on the line, usually the origin (0,0) for simplicity. - \(3(0)-2(0)>-13\) - \(0>-13\) which is true, so shade the region containing the origin. For inequality \(-x + 2y > 5\), test a point not on the line, for example, (0,0). - \(-0+2(0)>5\) - \(0>5\) which is false, so shade in the region not containing the origin.

Step 4: Identify the intersection of the shaded regions.

Finally, look at the graph and identify the intersection of the shaded regions. The intersection is the part of the graph where both inequalities are satisfied, and it represents the solution set of the system.

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

Look at the intersection of the shaded regions. If the intersection forms a closed region (such as a triangle, rectangle, etc.), it is bounded. If it extends infinitely in any direction, it is unbounded. After completing these steps, you will have graphically determined the solution set for the system of inequalities and identified whether it is bounded or unbounded.

Key Concepts

System of Inequalities

A system of inequalities consists of two or more inequalities that are considered simultaneously. It is similar to a system of equations, but instead of solving for a single point, a system of inequalities seeks regions on a graph where the inequalities overlap. When dealing with such systems, each inequality represents a condition that needs to be satisfied by the variables involved. For example, in the original exercise, we are given two inequalities:

• \(3x - 2y > -13\)


• \(-x + 2y > 5\)

The solution to the system of inequalities is the region on the coordinate plane where the solutions to each individual inequality coincide.Visualizing systems of inequalities graphically can make it easier to see all possible solutions that satisfy each condition at the same time. The overlapping region of the shaded areas for each inequality shows all possible solutions.

Bounded and Unbounded Regions

Identifying whether a solution set is bounded or unbounded is a crucial part of solving a system of inequalities. A bounded region is closed and finite. It does not stretch out to infinity and is typically contained within a geometric shape such as a triangle or rectangle. For example, if the intersection of shaded areas on a graph creates a shape with defined edges on all sides, it is considered bounded. On the other hand, an unbounded region extends infinitely in at least one direction. It represents solutions that satisfy the system of inequalities but do not reside within a confined space. An unbounded region might appear when there are fewer constraints or the constraints open up to infinity. In practice, determining if the solution set is bounded or unbounded involves looking at the graph and checking whether the overlap offers a complete enclosure or stretches endlessly.

Coordinate Plane Graphing

Graphing on a coordinate plane is the process of visually representing equations and inequalities by plotting points, lines, and shaded areas. A coordinate plane consists of two perpendicular number lines called axes: the horizontal line (x-axis) and the vertical one (y-axis). Coordinate plane graphing is essential when solving systems of inequalities. The steps for graphing involve:

• Rewriting each inequality as an equation to find the boundary line.


• Determining the intercepts (where the lines cross the axes) to plot the lines.


• Shading the region of the plane that satisfies the inequality. Often, the test point (like the origin) helps decide which side of the line to shade.

Graphical solutions provide a clear interpretation of the relationships between various inequalities by visualizing the overlap and intersections of shaded regions. This method illustrates how multiple conditions interact on the coordinate plane, highlighting the solution set for the system.