Chapter 11, Problem 86RE

To determine

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

Answer to Problem 86RE

  $\boxed{Unbounded}$

Explanation of Solution

Given information:

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

  $\{\begin{array}{l}x\ge 0\\ y\ge 0\\ 3x+y\ge 6\\ 2x+y\ge 2\end{array}$

Calculation:

Consider the sytem of inequalities.

  $\{\begin{array}{l}x\ge 0\\ y\ge 0\\ 3x+y\ge 6\\ 2x+y\ge 2\end{array}$

Let take inequality,

  $3x+y=6$

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,

  $\begin{array}{l}3\left(0\right)+\left(0\right)\ge 6\\ 0\ge 6\end{array}$

Since $0\ge 6$ is false statement point $\left(0,0\right)$ does not satisfying the inequality.

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

Now consider,

  $2x+y=2$

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

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

Since $0\ge 2$ is false statement point $\left(0,0\right)$ does not satisfying the inequality.

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

So the solution of all inequalities are given by the point in the region enclosed by lines and the graph is sketched in the rectangular Cartesian coordinate system as given below.

The graph extends to the infinity in both the perpendicular directions.

Hence, the graph of inequalities $\boxed{Unbounded}$ .