Answer
The graph is shown below:
Work Step by Step
Let us consider the inequalities $\begin{align}
& 3x+2y\ge 6 \\
& 2x+y\ge 6
\end{align}$
Substitute the equals symbol in place of the inequality and rewrite the equation as given below:
$\begin{align}
& 3x+2y=6 \\
& 2x+y=6
\end{align}$
To find the value of the x-intercept, substitute y = 0 as given below:
$\begin{align}
& 3x+2y=6 \\
& 3x+2\times 0=6 \\
& 3x=6 \\
& x=2
\end{align}$
To find the value of the y-intercept, substitute x = 0 as given below:
$\begin{align}
& 3x+2y=6 \\
& 3\times 0+2y=6 \\
& 2y=6 \\
& y=3
\end{align}$
To find another value of the x-intercept, substitute y = 0 as given below:
$\begin{align}
& 2x+y=6 \\
& 2x+0=6 \\
& 2x=6 \\
& x=3
\end{align}$
To find another value of the y-intercept, substitute x = 0 as given below:
$\begin{align}
& 2x+y=6 \\
& \left( 2\times 0 \right)+y=6 \\
& y=6
\end{align}$
Then, take origin $\left( 0,0 \right)$ as a test point and check the region in the graph to shade:
$\begin{align}
& 3x+2y\ge 6\text{ and }2x+y\ge 6 \\
& 0\ge 6\text{ and }0\ge 6 \\
\end{align}$
As we see that both above expressions are incorrect, we know that the shaded region will not contain the test point; that is, the region away from the origin must be shaded.
Thus, the graph of the given inequality is plotted and the solution set lies in the shaded region.
