Problem 33 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

The vertices are (0,0), (0,7), (5,2), and (7,0). The solution set is bounded.

Step 1: Identify the inequalities

We have the following inequalities to work with:1. \( x \geq 0 \)2. \( y \geq 0 \)3. \( x \leq 5 \)4. \( x + y \leq 7 \)

Step 2: Graph the non-negativity constraints

Since \(x \geq 0\) and \(y \geq 0\), the solution is limited to the first quadrant, meaning only points where both \(x\) and \(y\) are non-negative will be considered.

Step 3: Graph the inequality \(x \leq 5\)

Draw a vertical line at \(x = 5\). The solution set includes all points to the left of this line, within the first quadrant.

Step 4: Graph the inequality \(x + y \leq 7\)

First, plot the line \(x + y = 7\) by finding intercepts: at \(x = 0\), \(y = 7\), and at \(y = 0\), \(x = 7\). Shade the area below this line in the first quadrant, which includes the origin (0,0).

Step 5: Find the vertices

The vertices are at intersections of the boundary lines of the inequalities:- Intersect \(x = 0\) and \(x + y = 7\): (0, 7)- Intersect \(y = 0\) and \(x + y = 7\): (7, 0)- Intersect \(x = 5\) and \(x + y = 7\): (5, 2)- Intersect \(x = 5\) and \(y = 0\): (5, 0)Thus, the vertices of the solution region are (0,0), (0,7), (5,2), and (7,0).

Step 6: Determine if the solution set is bounded

The solution set is bounded since the region within the graph results a closed shape enclosed by the lines \(x = 0\), \(y = 0\), \(x = 5\), and \(x+y = 7\). It forms a trapezoid, having definite vertices.

Key Concepts

Graphing Inequalities

In mathematics, graphing inequalities helps visually represent the solutions of an inequality or a system of inequalities on a coordinate plane. This is especially useful as it allows us to easily understand how different conditions shape the possible solutions.

• First, consider simple inequalities like \( x \geq 0 \) and \( y \geq 0 \). These establish constraints where only non-negative values of \( x \) and \( y \) are considered, confining our solution to the first quadrant of the Cartesian plane.
• Next, we have \( x \leq 5 \). To graph this, draw a vertical line where \( x = 5 \) on the plane. The solution includes points to the left of this line. Here, because of the previous constraints, we consider only those in the first quadrant.
• The inequality \( x + y \leq 7 \) introduces a more dynamic boundary. Graph this by plotting points where the sum of \( x \) and \( y \) equals 7. Typically, intercepts such as \( (0, 7) \) and \( (7, 0) \) help mark this line. Points beneath this line within the bounded area satisfy the inequality.

Creating a systematic graph with these steps reveals a shaded region showing all possible solutions that satisfy each inequality simultaneously.

Vertices of a Polygon

When graphing systems of inequalities, the points where the boundary lines intersect form what's known as the vertices of the polygon. Identifying these vertices is crucial as they define the shape and extent of the solution area.

• The vertices for the inequalities \( x \geq 0 \), \( y \geq 0 \), \( x \leq 5 \), and \( x + y \leq 7 \) are identified by finding where each pair of lines intersects.
• By inspecting intersections, we find:
  • Where \( x = 0 \) meets \( x + y = 7 \): the vertex is \( (0, 7) \).
  • Where \( y = 0 \) meets \( x + y = 7 \): the vertex is \( (7, 0) \).
  • Where \( x = 5 \) meets \( x + y = 7 \): the vertex is \( (5, 2) \).
  • Where \( x = 5 \) meets \( y = 0 \): the vertex is \( (5, 0) \).

These vertices—\( (0,0) \), \( (0,7) \), \( (5,2) \), and \( (7,0) \)—can be connected to form a polygon. This bounded area contains all the solutions to the system.

Bounded Solution Set

Understanding whether a solution set is bounded or unbounded is vital when solving systems of inequalities. A bounded solution set is enclosed on all sides, meaning it forms a closed, finite shape on the graph.

• For the given system, the solution set forms a trapezoid within the coordinate plane.
• The boundaries are defined by the lines \( x = 0 \), \( y = 0 \), \( x = 5 \), and \( x+y = 7 \). This enclosing forms a polygon, thus confirming that the set is bounded.

Each boundary contributes to the enclosed region, indicating that solutions exist within a finite and closed area. Such regions are crucial for practical applications, allowing us to understand the limits and extent of possible values. This bounded set clearly defines where all constraints of the inequalities hold true, ensuring solutions adhere to the given limits.