Problem 25 - Algebra and Trigonometry

Question

21-46 . Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$ \left\\{\begin{array}{l}{y \leq-2 x+8} \\ {y \leq-\frac{1}{2} x+5} \\ {x \geq 0, \quad y \geq 0}\end{array}\right. $$

Step-by-Step Solution

Answer

The vertices of the solution set are (0, 5), (0, 8), (2, 4), and (4, 0). It is bounded.

Step 1: Interpret the Inequalities

Let's interpret each inequality given in the system. We have four constraints: 1. \( y \leq -2x + 8 \) 2. \( y \leq -\frac{1}{2}x + 5 \) 3. \( x \geq 0 \) 4. \( y \geq 0 \). This tells us that the solution set will be determined by the regions below the lines of these two inequalities, limited by the axes.

Step 2: Graph the Boundary Lines

Graph the boundary equation \( y = -2x + 8 \) by plotting points. Use points like \((0, 8)\) and \((4, 0)\). Next, graph \( y = -\frac{1}{2}x + 5 \) with points such as \((0, 5)\) and \((10, 0)\). These are dashed because the inequalities are \( \leq \). Also graph the axes \( x = 0 \) and \( y = 0 \) (solid lines).

Step 3: Determine Feasible Region

The feasible region is where all inequalities overlap. This will be the area below both boundary lines and above the x-axis, in the first quadrant (\( x \geq 0 \) and \( y \geq 0 \)). Shade this region.

Step 4: Find Intersection Points

To find the vertices of the feasible region, solve for intersection points. For where \( y = -2x + 8 \) and \( y = -\frac{1}{2}x + 5 \): Set \(-2x + 8 = -\frac{1}{2}x + 5\) and solve for \( x \). This yields \( x = 2 \). Substitute \( x = 2 \) into \( y = -2x + 8 \) to find \( y = 4 \). So, there is an intersection at \((2, 4)\).

Step 5: Identify Additional Vertices

Other vertices occur where the lines intersect with the axes: \((0, 8)\) for \( y = -2x + 8 \), \((0, 5)\) for \( y = -\frac{1}{2}x + 5 \), and because the inequalities are bound to the first quadrant, end vertices on the axes are \((4, 0)\).

Step 6: List All Vertices and Conclusion

List the vertices of the feasible region: \((0, 5)\), \((0, 8)\), \((2, 4)\), and \((4, 0)\). Determine boundedness: Check if the feasible region is closed or open. Here, it is closed and bounded as it forms a polygon within the axis constraints and intersecting lines.

Key Concepts

System of Inequalities

When dealing with a system of inequalities, we are essentially solving multiple inequalities at once. These inequalities define conditions over areas in the coordinate plane. Our goal is to find a common region that satisfies all these inequalities simultaneously. In this exercise, we have four key inequalities:

• \(y \leq -2x + 8\)
• \(y \leq -\frac{1}{2}x + 5\)
• \(x \geq 0\)
• \(y \geq 0\)

These inequalities tell us to focus on the areas below the given lines and within the first quadrant of the coordinate plane, restricted by the x and y axes.

Feasible Region

The feasible region is an important concept in graphing inequalities. It represents the area where all conditions of the inequalities overlap. We need to find where each inequality's shaded region intersects with the others. This overlapping region is our feasible region. It is essential because any solution to our system of inequalities must lie in this area.

In this problem, the feasible region is confined to the first quadrant due to the conditions \(x \geq 0\) and \(y \geq 0\). This adds a natural boundary along the x and y axes. The feasible region can often be visualized as the part of the graph where all shadings from the inequalities coincide.

Intersection Points

Finding intersection points is crucial to pinpoint the corners, or vertices, of the feasible region. The intersection points occur where the boundary lines of the inequalities cross each other. These intersections give us precise guidance on where the feasible region is absolutely constrained.

To find these intersection points, we set the equations of the boundary lines equal and solve them algebraically. For example, to find the intersection of \(y = -2x + 8\) and \(y = -\frac{1}{2}x + 5\), we equate and solve:\[ -2x + 8 = -\frac{1}{2}x + 5 \]Solving gives us \(x = 2\), and by substituting back, we get \(y = 4\). Therefore, one intersection point is at \((2, 4)\). Such points encapsulate the parameters of the feasible region.

Bounded Solution Set

Whether a solution set is bounded or not refers to whether the feasible region is limited in all directions. A bounded region is contained entirely within a finite area. In this exercise, the solution set is indeed bounded.

The feasible region forms a closed polygon that is constrained by the given lines and the coordinate axes. It contains all the vertices

• \((0, 5)\)
• \((0, 8)\)
• \((2, 4)\)
• \((4, 0)\)

These vertices define the boundaries of the polygon, which makes sure the solution set does not extend indefinitely in any direction. Therefore, the bounded solution set ensures that there is a limit to the solutions we can have within the given constraints.