Problem 120 - Algebra and Trigonometry

Question

Exercises 120–122 will help you prepare for the material covered in the next section. a. Graph the solution set of the system: $$\left\\{\begin{array}{r} {x+y \geq 6} \\ {x \leq 8} \\ {y \geq 5} \end{array}\right.$$ 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 feasible region formed by the system of inequalities are (8,5), (8,6) and (1,5). The value of \(3x + 2y\) evaluated at these points are 34, 36 and 13, respectively.

Step 1: Graph the Inequalities

First, each of the inequalities 'x + y >= 6', 'x <= 8', 'y >= 5' is to be represented on a graph. It should be noted that the inequality 'x + y >= 6' represents all points above the line \(x + y = 6\), 'x <= 8' represents all points to the left of the line \(x = 8\) and 'y >= 5' includes all points above the line \(y = 5\). The area where all these inequalities satisfy forms the feasible region of the system.

Step 2: Identify the Corners of the Feasible Region

Corner points of the feasible region are the intersection points of the inequalities. In this case, the inequalities intersect at the points (8,5), (8,6) and (1,5). These are the corner points of the feasible region.

Step 3: Evaluate the Expression at Corner Points

The next step is to evaluate the expression \(3x + 2y\) at the corner points obtained. When x = 8, y = 5, the expression becomes \(3*8 + 2*5 = 34\). When x = 8, y = 6, it becomes \(3*8 + 2*6 = 36\). When x = 1, y = 5, it becomes \(3*1 + 2*5 = 13\).

Key Concepts

Graphing Inequalities

When graphed, inequalities like these will shade an area on the graph. The boundary of the shaded region is often defined by a straight line, and the type of line can vary. For inequalities such as \(\geq\) or \(\leq\), the line itself is included in the solution set and is typically drawn as a solid line. However, for strict inequalities \(>\) or \(<\), the line is not part of the solution and is often represented as a dashed line.

In the provided exercise, there are three inequalities: \(x + y \geq 6\), \(x \leq 8\), and \(y \geq 5\). Each is plotted on a graph as follows:

• The line \(x + y = 6\) is solid because the inequality is \(\geq\). Everything above this line is shaded.
• The line \(x = 8\) is also solid, and all points to the left are shaded since \(x \leq 8\).
• The line \(y = 5\) again uses a solid line with shading above due to \(y \geq 5\).

By assessing the shaded regions, you can determine the solution set where all the conditions of the inequalities overlap.

Feasible Region

The feasible region refers to the set of all possible points that satisfy all given inequalities. In a system of inequalities, this region can often be visualized as a polygonal area bounded by the intersecting lines of the graph.

For the exercise, the feasible region is the area on the graph where the three individual solutions of the inequalities overlap. This is the area that is simultaneously shaded by all inequalities involved. It is important to understand that the feasible region can be empty, a single point, or more commonly a polygonal shape depending on the constraints given by the inequalities.

To identify it correctly, observe where the shading occurs from each inequality in the system and focus on the common overlapping section. This intersection results in the feasible region, representing all potential solutions that meet the conditions set by all inequalities involved.

Corner Points

Corner points are the vertices of the feasible region. These points occur where the boundary lines of the inequalities intersect each other. Finding these points is crucial because they often represent potential optimal solutions, especially in problems dealing with maximization or minimization of a given expression.

In the exercise, the inequalities intersect at three important points: (8, 5), (8, 6), and (1, 5). These are the defined corner points of the feasible region.

• (8, 5) is where \(x = 8\) intersects \(y = 5\).
• (8, 6) results from \(x = 8\) and the line \(x + y = 6\).
• (1, 5) is the intersection of \(y = 5\) with the line \(x + y = 6\).

The evaluation of expressions like \(3x + 2y\) at these corner points is often explored to determine which point provides an optimal solution based on the context and criteria of a given problem.