Answer
The graph is shown below:
Work Step by Step
Let us consider the given inequalities $\begin{align}
& 3x+y\le 9 \\
& 2x+3y\ge 6 \\
& x\ge 0 \\
& y\ge 0
\end{align}$
Put the equals symbol in place of the inequality and rewrite the equation as given below:
By finding any two solutions of the linear equation, plot the graph of the linear equation $3x+y=9$:
To find the value of the x-intercept, put the value of y = 0 as given below:
$\begin{align}
& 3x+0=9 \\
& x=\frac{9}{3} \\
& x=3
\end{align}$
To find the value of the y-intercept, put the value of x = 0 as given below:
$\begin{align}
& 3\left( 0 \right)+y=9 \\
& y=9
\end{align}$
Plot the intercepts $\left( 3,0 \right)\text{ and }\left( 0,9 \right)$ and draw a solid line passing through these points. In the inequality there is the $\le $ symbol in which the equality is included.
Now, this solid line divides the plane in three regions -- the line itself and two half-planes.
Now, take the origin $\left( 0,0 \right)$ as a test point and check the region in the graph to shade:
Check if the test point satisfies the inequality.
$\begin{align}
3x+y\le 9 & \\
3\left( 0 \right)+0\overset{?}{\mathop{\le }}\,9 & \\
0\le 9 & \\
\end{align}$
Since the test point satisfies the inequality, shade the half-plane containing that test point towards the origin. Plot the graph using the intercepts as given below:
By finding any two solutions of the linear equation, plot the graph of the linear equation $2x+3y\ge 6$:
To find the value of the x-intercept, put the value of y = 0 as given below:
$\begin{align}
& 2x+3\left( 0 \right)=6 \\
& 2x=6 \\
& x=\frac{6}{2} \\
& x=3
\end{align}$
To find the value of the y-intercept, put the value of x = 0 as given below:
$\begin{align}
& 2\left( 0 \right)+3y=6 \\
& 3y=6 \\
& y=\frac{6}{3} \\
& y=2
\end{align}$
Plot the intercepts $\left( 3,0 \right)\text{ and }\left( 0,2 \right)$ and draw a solid line passing through these points; since the inequality contains the $\le $ symbol, the equality is included.
Now, this dashed line divides the plane in 3 region: the line itself, and the two half planes.
Then, take the origin $\left( 0,0 \right)$ as a test point and check the region in the graph to shade:
Check if the test point satisfies the inequality.
$\begin{align}
& 2x+3y\ge 6 \\
& 0+0\overset{?}{\mathop{>}}\,9 \\
& 0<9 \\
\end{align}$
Since, the test point does not satisfy the inequality, shade the half-plane not containing that test point that is away from the origin.
And to plot the equation $ x\ge 0$, graph the line $ x=0$ and shade the right part of the line; that is, shade the region for which x values will be positive, as the inequality contains the $\ge $ symbol.
Similarly, to plot the inequality $ y\ge 0$, graph the line $ y=0$ and shade the right part above the line; that is, shade the region for which y values will be positive, as the inequality contains the $\ge $ symbol.
See the final graph below.
