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

$\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\le 4\\ 2x+3y\le 6\end{array}\right.$

The given system of inequality is

$\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\le 4\\ 2x+3y\le 6\end{array}\right.$

Graph of $x=0\&y=0$ will be y-axis and x-axis respectively

So region right side of they-axis and above thex-axis will be shaded

Graph of $x+y=4$will be a line

now test the point $(0,0)$for inequality$x+y\le 4$

$$\begin{gathered} x+y\le 4 \\ 0=0\le 4 \\ 0<4 \end{gathered}$$

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

So the region toward origin from the line will be shaded

now test the origin point $(0,0)$for inequality$2x+3y\le 6$

$$\begin{gathered} 2x+3y\le 6 \\ 2\left(0\right)+3\left(0\right)\le 6 \\ 0<6 \end{gathered}$$

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

So the region toward origin from the line will be shaded

To plot the graph of $\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\le 4\\ 2x+3y\le 6\end{array}\right.$, Plot a solid line $x+y=4$and shade the region towards origin from the line, plot a solid line $2x+3y=6$ and shade the region towards origin from the line and plot they-axis and x-axis and shade the region right side of the y-axis and above the x-axis

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

Boundry lines of inequalities intersect at the points $(0,0),(0,2),\&(3,0)$so corner points are$(0,0),(0,2),\&(3,0)$