Problem 22 - 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. $$ \begin{array}{rr} x+y & \geq-2 \\ 3 x-y & \leq 6 \end{array} $$

Step-by-Step Solution

Answer

The solution set is an unbounded region represented by the overlap of the shaded regions satisfying both inequalities \(x+y\geq-2\) and \(3x-y\leq6\).

Step 1: Graph the inequality lines

To graph the inequality lines, we first rewrite them as equalities: \(x+y=-2\) and \(3x-y=6\) Now, find two points on each line that we can use to plot the line. We will use the x- and y-intercepts as these points. For the first line, \(x+y=-2\): x-intercept (when y=0): \(x=-2\) y-intercept (when x=0): \(y=-2\) Points: \((-2,0),(0,-2)\) For the second line, \(3x-y=6\): x-intercept (when y=0): \(x=2\) y-intercept (when x=0): \(y=-6\) Points: \((2,0),(0,-6)\) Plot these two lines on the coordinate plane.

Step 2: Identify the shaded region

Now we need to determine which side of the lines represents the solution to the inequalities. For this, we can choose a test point outside the solution region, usually (0,0) works best unless it lies on any of the lines. In this case, (0,0) is not on any of the lines, so we can use it as our test point. For the first inequality, \(x+y\geq-2\): \(0+0\geq-2\) \(0\geq-2\) - True, so we shade the region on the same side as (0,0). For the second inequality, \(3x-y\leq6\): \(3(0)-(0)\leq6\) \(0\leq6\) - True, so we shade the region on the same side as (0,0). The solution set is the overlap of the two shaded regions.

Step 3: Identify if the solution set is bounded or unbounded

Examine the graph to determine if the solution region has any boundaries. In this case, the shaded region is not enclosed by the lines, and it extends outward indefinitely. Thus, the solution set is unbounded.

Step 4: Final Answer

The solution set is an unbounded region represented by the overlap of the shaded regions satisfying both inequalities \(x+y\geq-2\) and \(3x-y\leq6\).

Key Concepts

Inequalities

Inequalities are mathematical expressions that show the relationship between two values, indicating that one is either greater than, less than, or equal to the other. They are represented using symbols such as \( >, <, \geq, \leq \). These expressions describe ranges of possible solutions rather than specific numbers. For example, an inequality like \(x + y \geq -2\) means that the sum of \(x\) and \(y\) should be greater than or equal to \(-2\).
Understanding inequalities is crucial because they allow us to express conditions and constraints in a wide range of mathematical problems, including optimization, economics, and real-world scenarios.
To solve a system of inequalities graphically, we translate these mathematical conditions into visual representations on a coordinate plane. This helps us find a set of solutions that satisfy all given inequalities at once.

• The symbolic representation helps to define which side of a boundary line contains solutions.
• Combining different inequalities can result in a shared region where all conditions are met.

Grasping these concepts allows students to systematically find and interpret solutions to more complex mathematical scenarios.

Graphing Inequalities

Graphing inequalities involves plotting them on a coordinate plane to find the set of solutions that satisfy the inequality. This process helps visualize the inequalities as regions on the graph. Let's break it down into steps:
First, change the inequality to an equality to identify its boundary line, such as transforming \(x + y \geq -2\) into \(x + y = -2\). Find the x- and y-intercepts, which are the points where the line crosses the x-axis and y-axis, respectively. Plot these points and draw the line.

• For the line \(x + y = -2\), the x-intercept is \((-2, 0)\) and y-intercept is \((0, -2)\).
• For the line \(3x - y = 6\), the x-intercept is \((2, 0)\) and y-intercept is \((0, -6)\).

Next, select a test point, preferably (0,0), to determine which side of each line the solution set lies on. If the test point satisfies the inequality, shade the region that includes the test point on the graph.
The shaded region that represents the solution set of a system of inequalities is where the shaded areas of all individual inequalities overlap. This overlap represents all the points that simultaneously satisfy all inequalities.

Unbounded Solution Sets

An unbounded solution set within the context of graphically solving systems of inequalities refers to the regions on the graph that extend infinitely in one or more directions. This implies there is no limit to the values that \(x\) and \(y\) can take as long as they satisfy the inequalities.
When analyzing the solution set for the inequalities \(x+y \geq -2\) and \(3x-y \leq 6\), we notice that the overlap of the shaded regions creates an area that is not enclosed within finite boundaries. This means the region continues indefinitely without being trapped or limited by the lines drawn on the graph.

• If the shaded regions extend beyond any direction without being bounded by another line, it's considered an unbounded solution set.
• This occurs when the lines do not form a closed figure around the shaded region.

Recognizing whether a solution is bounded or unbounded is essential for understanding the scope and limitations of possible solutions in real-world applications. It signifies whether there are constraints on the values \(x\) or \(y\) can take or if they are open-ended.