Chapter 11, Problem 84RE

To determine

Graph each system of inequalities. Verify whether the graph is bounded or unbounded.

Answer to Problem 84RE

  $\boxed{unbounded}$

Explanation of Solution

Given information:

Graph each system of inequalities. Tell whether the graph is bounded or unbounded, and label the corner points.

  $\{\begin{array}{l}x-2y\le 6\\ 2x+y\ge 2\end{array}$

Calculation:

Consider the inequality.

  $\{\begin{array}{l}x-2y\le 6\\ 2x+y\ge 2\end{array}$

To graph the given inequality, start by graphing the lines $x-2y=6$ and $2x+y=2$ . To determine the part of the graph to be shaded use a test point $\left(0,0\right)$ .

Substitute $x=0,y=0$ in the first inequality of the given system,

  $\begin{array}{l}\left(0\right)-2\left(0\right)\le 6\\ 0\le 6\end{array}$

since $0\le 6$ satisfy the inequality at point $\left(0,0\right)$

Hence the complete solution is set of all points which are same side of the line as the point $\left(0,0\right)$ .

Now consider,

  $2x+y=2$ Use a test point $\left(0,0\right)$ .

Substitute $x=0,y=0$ in the first inequality of the given system,

  $\begin{array}{l}2\left(0\right)+\left(0\right)\ge 2\\ 0\ge 2\end{array}$

  $0\ge 2$ is false statement ,

Hence the complete solution is set of all points which are not at same side of the line as the point $\left(0,0\right)$

The graph of the given system,

Hence, the graph of the given inequalities extends to the infinity in both the perpendicular directions so that it is $\boxed{unbounded}$ .