Problem 33 - Algebra and Trigonometry

Question

19–40 Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{aligned} x & \geq 0 \\ y & \geq 0 \\ x & \leq 5 \\ x+y & \leq 7 \end{aligned}\right.$$

Step-by-Step Solution

Answer

Vertices: (0, 0), (0, 7), (2, 5), (5, 0). Solution set is bounded.

Step 1: Identify the inequalities

The system of inequalities consists of four constraints: \(x \geq 0\), \(y \geq 0\), \(x \leq 5\), and \(x + y \leq 7\). These inequalities define a region in the xy-plane.

Step 2: Graph the inequalities

Begin by graphing each inequality on the coordinate plane. Start with \(x \geq 0\) which is the region to the right of the y-axis. Then graph \(y \geq 0\) which is the region above the x-axis. Next, graph \(x \leq 5\) which is the region to the left of the vertical line \(x = 5\). Finally, graph \(x + y \leq 7\) by plotting the line \(x + y = 7\) and shading below it. The intersection of these regions represents the solution set.

Step 3: Find the vertices of the solution set

To find the vertices, determine the points of intersection of the boundary lines. The vertices of the region are where these lines intersect; calculate or analyze where these boundary lines intersect to get the vertices: (0, 0), (0, 7), (2, 5), and (5, 0).

Step 4: Determine if the solution set is bounded

A solution set is bounded if it forms a closed and finite region. In this case, since the intersection of the inequalities results in a quadrilateral with vertices (0, 0), (0, 7), (2, 5), and (5, 0), the solution set is indeed bounded as it is enclosed within these points.

Key Concepts

Graphing Inequalities

When dealing with systems of inequalities, a key step is graphing each inequality individually on a coordinate plane. This allows us to visually identify the solution set, which is the area where all inequalities overlap. Here's how you can approach it:

• Start with simple geometric constraints like \(x \geq 0\) and \(y \geq 0\). These inequalities indicate that the solution set will be in the first quadrant of the Cartesian plane, where both \(x\) and \(y\) are positive.


• Next, consider inequalities such as \(x \leq 5\). This can be visualized as a vertical line moving towards the left of the plane from 5, further constraining the region on the plane.


• Finally, for an inequality like \(x + y \leq 7\), plot the line \(x + y = 7\) and shade below it to indicate the region that meets the inequality criteria.

By graphing these, you can see the collective area that satisfies all conditions, making it easier to solve problems that rely on multiple constraints.

Bounded Solution Set

A bounded solution set occurs when the intersection of inequality regions creates a finite and closed-off area. In the exercise's context, the combination of the inequalities \(x \geq 0\), \(y \geq 0\), \(x \leq 5\), and \(x+y \leq 7\) yields a shape bound by these constraints.

This specific scenario leads to a quadrilateral on the coordinate plane, indicating a bounded region. This quadrilateral forms because each of these inequalities confines the solution set further:

• The vertical and horizontal lines (\(x = 0\) and \(y = 0\)) box it into the first quadrant.


• The line \(x = 5\) creates a clear left boundary.


• The diagonal line \(x + y = 7\) limits the maximum sum of x and y, thus finalizing the enclosure.

A bounded solution set is important as it suggests that a practical, limited set of solutions exists. This has immediate implications in applications where resources or conditions are restricted to a manageable domain.

Vertices of a Region

Vertices are the corner points of a region formed by the intersection of lines when graphing inequalities. These intersections are crucial because they define the precise edges of the solution area. Finding these vertices involves solving the pairs of the boundary lines where they intersect.

In the given system:

• Intersection of \(x = 0\) and \(y = 0\) gives the origin point of (0, 0).


• \(x = 0\) and \(x + y = 7\) intersect at (0, 7).


• \(x = 5\) and \(x + y = 7\) intersect at (5, 0).


• Finally, solving for \(x + y = 7\) and \(x = 5\) results in the point (2, 5).

These four vertices (0, 0), (0, 7), (2, 5), and (5, 0) collectively outline the bounded region. Highlighting vertices clearly demonstrates how multiple constraints come together to form a unified solution area, allowing for precise mathematical modeling and solutions.