Answer
The graph is shown below:
Work Step by Step
Let us consider the inequality $3x-4y\gt12$; substitute the equal symbol in place of the inequality and rewrite the equation as given below:
$3x-4y=12$
To find the value of the x-intercept, put y = 0 as given below:
$\begin{align}
& 3x-4y=12 \\
& 3x-\left( 4\times 0 \right)=12 \\
& 3x=12 \\
& x=4
\end{align}$
To find the value of the y-intercept, substitute x = 0 as given below:
$\begin{align}
& 3x-4y=12 \\
& \left( 3\times 0 \right)-4y=12 \\
& -4y=12 \\
& y=-3
\end{align}$
Then, take the origin $\left( 0,0 \right)$ as a test point and check the region in the graph to shade:
$\begin{align}
& 3x-4y>12 \\
& \left( 3\times 0 \right)-\left( 4\times 0 \right)>12 \\
& 0>12
\end{align}$
So, the above expression is incorrect, which means the shaded region will not contain the test point; that is, the region away from the origin will be shaded. Therefore, the line passes through points $\left( 4,0 \right)$ and $\left( 0,-3 \right)$.
Plot the graph using the intercepts as given below.
Thus, the graph of the provided inequality is plotted and the line passes through points $\left( 4,0 \right)$ and $\left( 0,-3 \right)$. Also, we get that the shaded region does not contain the origin.
