Problem 37 - 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. $$ \left\\{\begin{array}{c}{x+2 y \leq 14} \\ {3 x-y \geq 0} \\ {x-y \geq 2}\end{array}\right. $$

Step-by-Step Solution

Answer

Vertices are at \((2, 6)\), \((-1, -3)\), and \((6, 4)\). The region is bounded.

Step 1: Write the Inequalities

Let's start by writing out the inequalities given in the exercise:1. \(x + 2y \leq 14\)2. \(3x - y \geq 0\)3. \(x - y \geq 2\)

Step 2: Solve Each Inequality for y

Rearrange each inequality to solve for \(y\) in terms of \(x\):1. \(x + 2y \leq 14\) becomes \(2y \leq -x + 14\), so \(y \leq -\frac{1}{2}x + 7\).2. \(3x - y \geq 0\) gives us \(y \leq 3x\).3. \(x - y \geq 2\) becomes \(y \leq x - 2\).

Step 3: Graph the Lines

Graph each line derived from the solution for \(y\):1. Graph the line \(y = -\frac{1}{2}x + 7\).2. Graph the line \(y = 3x\).3. Graph the line \(y = x - 2\).Use dashed lines for inequalities.

Step 4: Determine the Feasible Region

Shade the region that satisfies all three inequalities:- Below (or on) the line \(y = -\frac{1}{2}x + 7\).- Below (or on) the line \(y = 3x\).- Below (or on) the line \(y = x - 2\).The feasible region is the intersection of all these areas.

Step 5: Find the Vertices of the Region

To find the vertices, identify the points of intersection between pairs of lines:1. Intersection of \(y = -\frac{1}{2}x + 7\) and \(y = 3x\): \[-\frac{1}{2}x + 7 = 3x\] \[7 = \frac{7}{2}x\] \[x = 2, \ y = 6\].2. Intersection of \(y = 3x\) and \(y = x - 2\): \[3x = x - 2\] \[2x = -2\] \[x = -1, \ y = -3\].3. Intersection of \(y = -\frac{1}{2}x + 7\) and \(y = x - 2\): \[-\frac{1}{2}x + 7 = x - 2\] \[7 + 2 = x + \frac{1}{2}x\] \[9 = \frac{3}{2}x\] \[x = 6, \ y = 4\].

Step 6: Assess if the Region is Bounded

Determine whether the feasible region is bounded. The region does not extend infinitely in any direction, hence it is bounded.

Key Concepts

system of inequalities

A system of inequalities consists of two or more inequalities that are considered simultaneously. In this particular exercise, we have the following system of inequalities:

• \(x + 2y \leq 14\)
• \(3x - y \geq 0\)
• \(x - y \geq 2\)

Graphing these inequalities involves plotting the solutions that satisfy all conditions at once. This means each region that we shade for one inequality must overlap with the shaded regions of the others, indicating that all points in the intersection zone meet the criteria set by the system. Understanding a system of inequalities is crucial when modeling situations where multiple conditions must be satisfied simultaneously.

bounded solution

A bounded solution refers to a region that is contained within some finite boundaries, without extending to infinity in any direction. It is a closed and limited area on the graph.
In this exercise, the feasible region formed by the system of inequalities is bounded. This is determined by observing that the region doesn't extend outwards indefinitely. Instead, it is enclosed by the lines created by each inequality. A bounded solution is different from an unbounded one, where the feasible region stretches endlessly, typically in one or more directions. Recognizing whether a solution is bounded or not is essential in understanding the constraints of the problem you are dealing with.

vertices coordinates

Vertices coordinates are the specific points where the boundary lines of the feasible region intersect. These coordinates define the corners of the feasible region on the graph. Finding the intersection of the lines helps determine these critical points:

• The intersection of \(y = -\frac{1}{2}x + 7\) and \(y = 3x\) gives the vertex at \((2, 6)\).
• The intersection of \(y = 3x\) and \(y = x - 2\) provides the vertex \((-1, -3)\).
• The intersection of \(y = -\frac{1}{2}x + 7\) and \(y = x - 2\) is the vertex \((6, 4)\).

These vertices help in evaluating the extremities of the applicable region. Identifying these points accurately is necessary for calculating the boundaries of the feasible area, crucial especially in optimization problems like linear programming.

feasible region

The feasible region is the overlapping area that satisfies all the inequalities in the system. It is the intersection of all regions obtained from each inequality. In the context of graphing inequalities, the feasible region is often shaded to highlight where all conditions hold true.
For our system, the shaded region lies:

• Below or on all of the lines defined by our inequalities.
• Includes points for which all inequalities are true.

Identifying the feasible region helps visualize constraints and optimize solutions within these constraints. In real-world situations, this region represents all possible solutions to the problem you're trying to solve under the given conditions.