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.

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

Short answer

The graph of $\left\{\begin{array}{l}-2x+y\le 2\\ x+y\ge 2\end{array}\right.$is unbounded and the corner point is $\left(0,2\right)$and the graph is

Step-by-step solution

Step 1. Given data 

The given system of inequality is

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

Step 2. Properties of the graph of the inequality 

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

now test the point $(0,0)$for inequality

localid="1646956260525" $$\begin{gathered} -2x+y\le 2 \\ -2\left(0\right)+\left(0\right)\le 2 \\ 0\le 2 \end{gathered}$$

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

So shade the region from line$-2x+y=2$toward the origin

Step 3. Properties of the graph of the inequality 

Graph of $x+y=2$will be lines

now test the point$(0,0)$for inequality

role="math" localid="1646956963236" $$\begin{gathered} x+y\ge 2 \\ 0+0\ge 2 \\ 0<2 \end{gathered}$$

inequality does not satisfy so the origin will not lie in the region of inequalityrole="math" localid="1646956562016" $x+y\ge 2$

So shade the region above the line$x+y=2$

Step 4. graph of the system of inequality   

Plot solid lines $-2x+y\le 2$and shade the region from line toward the origin and plot the line $x+y\ge 2$and shade the region away from origin to plot the graph of $\left\{\begin{array}{l}-2x+y\le 2\\ x+y\ge 2\end{array}\right.$

Graph extended infinitely in positive x and y direction so the graph is unbounded

Boundry lines of both inequalities intersect at the point $(0,2)$so corner point is$(0,2)$