Chapter 11, Problem 48RE

To determine

To graph: Consider the graph of inequalities $\left\{\begin{array}{c}x\ge 0\\ y\ge 0\\ 2x+y\le 8\\ x+2y\ge 2\end{array}\right\}$ .

Explanation of Solution

Given information:

The given inequalities are $\left\{\begin{array}{c}x\ge 0\\ y\ge 0\\ 2x+y\le 8\\ x+2y\ge 2\end{array}\right\}$ .

Calculation:

Consider the graph to be shaded, use a test point $\left(0,0\right)$ .

Substitute 0 for $x$ and $y$ in the first inequality of the system.

  $\begin{array}{c}2\left(0\right)+\left(0\right)\le 8\\ 0<8\end{array}$

The point $\left(0,0\right)$ satisfies the inequality $2x+y\le 8$ . So, we shade the region below the line.

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

The graph of inequality figure-(1) shows.

  

Figure-(1)

The graph of a system is said to be bounded if it can be contained within a circle of sufficiently large radius. If the graph cannot be contained, then it is said to be unbounded.

Label the corner points in the given graph. From the graph above, we can see that there corner points $\left(0,1\right)$ , $\left(0,8\right)$ , $\left(2,0\right)$ and $\left(4,0\right)$ .

The graph of inequality figure-(2) shown.

  

Figure-(2)

Therefore, the intersection of the shaded regions gives the graph of the system of inequalities.