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Chapter 3: Problem 35

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded. $$ \begin{aligned} x-y & \geq-6 \\ x-2 y & \leq-2 \\ x+2 y & \geq 6 \\ x-2 y & \geq-14 \\ x \geq 0, y & \geq 0 \end{aligned} $$

Short Answer

Expert verified
The solution set for the given system of inequalities is the overlapping region on the coordinate plane when graphing all inequalities. This region represents the values of x and y that satisfy all inequalities simultaneously. In this case, the solution set is a polygon, indicating a bounded solution set.

Step by step solution

01

Graph Each Inequality Separately

For this step, consider each inequality as a line by changing the inequality to an equation. Then, decide on which side of the line the solution of the inequality lies. \(x - y = -6\) The solution of this inequality lies above the line. \(x - 2y = -2\) The solution of this inequality lies below the line. \(x + 2y = 6\) The solution of this inequality lies above the line. \(x - 2y = -14\) The solution of this inequality lies above the line. \(x = 0\), The solution of this inequality lies to the right of the line. \(y = 0\), The solution of this inequality lies above the line.
02

Locate the Overlapping Region

To locate the overlapping region, graph all the lines from Step 1 on the same coordinate plane. Shade the regions representing the solution of each inequality so that the overlapping region becomes visible.
03

Determine if the Solution Set is Bounded or Unbounded

Examine the overlapping region. If the region is enclosed within a finite boundary, it is a bounded solution set. If the region extends infinitely in any direction, it is an unbounded solution set. To help decide if the solution set is bounded or unbounded, check if there is an enclosed polygon formed by the intersection points of the inequalities, or if any sides of the region are not limited by the inequalities. After performing these steps, the solution(set) to the given system of inequalities can be identified, along with whether it is bounded or unbounded.

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