Problem 116 - College Algebra

Question

Exercises 116-118 will help you prepare for the material covered in the next section. a. Graph the solution set of the system: $$\left\\{\begin{aligned} x+y & \geq 6 \\ x & \leq 8 \\ y & \geq 5 \end{aligned}\right.$$ b. List the points that form the corners of the graphed region in part (a). c. Evaluate \(3 x+2 y\) at each of the points obtained in part (b).

Step-by-Step Solution

Answer

The corner points of the region are (8,5), (8,3), and (3,5). When \(3x+2y\) is evaluated at these points, it results in \(38\), \(34\), and \(23\) respectively.

Step 1: Plotting the Inequalities

Start by graphing each inequality on the same set of axes. When graphing the inequality \(x + y \geq 6\), this line will be solid (because of the 'equal to' component in the inequality) and everything above the line will be shaded. For the inequality \(x \leq 8\), draw a vertical line at \(x = 8\) and shade everything to the left. For \(y \geq 5\), draw a horizontal line at \(y = 5\) and shade everything above.

Step 2: Identifying the Corners of the Region

The area shaded by all inequalities forms a convex polygon. The vertices of this polygon are the corner points, which essential in finding the maximum or minimum of a function defined over this region. For this particular setup, there are three corner points, at the coordinates (8,5), (8,3) and (3,5).

Step 3: Evaluating Function at Corner Points

Evaluate the given expression \(3x+2y\) at each of the corner points identified in the previous step. The results are as follows: At (8,5), the expression evaluates to \(38\). At (8,3), it evaluates to \(34\). At (3,5), it evaluates to \(23\).

Key Concepts

Understanding the Solution Set

The solution set of a system of inequalities represents all the values that satisfy every inequality equation in the system. In our exercise, we have three inequalities:

• \(x + y \geq 6\)
• \(x \leq 8\)
• \(y \geq 5\)

To find the solution set, begin by graphing each inequality on a coordinate plane.
When graphing, remember:

• Solid Lines: Use a solid line to represent inequalities with \(\geq\) or \(\leq\), as they include the boundary line itself.
• Shading: Shade the area that fulfills the inequality (above the line for \(\geq\) and below for \(\leq\)).

For this exercise, the intersection region of the shaded areas represents the solution set. It is crucial as it shows all the possible combinations of \(x\) and \(y\) that satisfy all three given conditions. Graphically, this region will appear as a shaded area on the graph, often forming a polygon.

Finding the Corner Points

Corner points, also known as vertices, are pivotal when evaluating the solution set of inequalities. These are the intersections of the boundary lines that describe the region representing the solution set. Think of them as crucial markers that outline the area where all the conditions meet.
To identify them:

• Look where the lines of the inequalities intersect.
• Note these intersections, as they are the corner points of the polygonal region.

For our exercise, we find that the region formed by intersecting three inequalities has the following corner points:

• (8,5)
• (8,3)
• (3,5)

These points outline the limits of possible solutions. They help in determining extrema values for expressions evaluated across the solution set. Knowing these specific coordinates is essential when looking to analyze the behavior of a function defined by the expression at these key points.

Evaluating Expressions at Given Points

Evaluating expressions at corner points is the next step once you've identified the solution set's boundary points. Here, you will calculate the value of a given expression using the coordinates of the corner points found in the graph.
The expression provided in the exercise is \(3x + 2y\). We evaluate this expression at each corner point:

• At (8,5): Substitute the values into the expression \(3 \times 8 + 2 \times 5 = 24 + 10 = 34\).
• At (8,3): Substitute the values \(3 \times 8 + 2 \times 3 = 24 + 6 = 30\).
• At (3,5): Substitute the values \(3 \times 3 + 2 \times 5 = 9 + 10 = 19\).

By calculating these values, you gain insight into how the expression behaves at these critical points of the solution set. This process is key for tasks like optimization, where determining maximum or minimum values is necessary. Understanding these steps fully is essential for mastering the technique of graphing systems of inequalities and evaluating functions.