Question

In Problems 46–48, graph each system of inequalities by hand. Tell whether the graph is bounded or unbounded, and label the corner points.

$\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ 2x+y\le 8\\ x+2y\ge 2\end{array}\right.$

Short answer

The graph of $\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ 2x+y\le 8\\ x+2y\ge 2\end{array}\right.$is bounded and the corner points are $(2,0),(4,0),(0,1),\&(0,8)$and the graph is

Step-by-step solution

Step 1. Given data   

The given system of inequality is

$\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ 2x+y\le 8\\ x+2y\ge 2\end{array}\right.$

Step 2. Properties of the graph of the inequality   

Graph of will be y-axis andx-axis respectively

So region right side of the y-axis and above the x-axis will be shaded

Graph of$2x+y=8$will be a line

now test the point for inequality$2x+y\le 8$

role="math" localid="1646960146459" $$\begin{gathered} 2x+y\le 8 \\ 2\left(0\right)+0\le 8 \\ 0<8 \end{gathered}$$

inequality satisfied so the origin will lie in the region of inequality

So the region toward origin from the line will be shaded

Step 3. Properties of the graph of the inequality   

Graph of $x+2y=2$will be a line

now test the origin point for inequality $x+2y\ge 2$

$$\begin{gathered} x+2y\ge 2 \\ 0+2\left(0\right)\ge 2 \\ 0<2 \end{gathered}$$

inequality does not satisfy so the origin will not lie in the region of inequality

So the region above from the line will be shaded

Step 4. graph of the system of inequality  

Graph of the system of inequality is

Graph of the system of inequalities state that the graph is bounded

Boundry lines of inequalities intersect at the points$(2,0),(4,0),(0,1),\&(0,8)$ so corner points are$(2,0),(4,0),(0,1),\&(0,8)$