The given system of inequalities is:
[tex]\begin{gathered} y>6x-4 \\ 3x+9y\leq18 \end{gathered}[/tex]
We can observe on the first inequality, it is in the slope-intercept form y>mx+b, where m is the slope and b is the y-intercept. So m=6 and b=-4.
Then, this inequality represents the line that intercepts the y-axis at -4.
Also, as this has the symbol ">", it means the line itself is not included in the solution, then it is a dotted line and its graph is:
Now, for the second inequality, we need to convert it to slope-intercept form:
[tex]\begin{gathered} 3x+9y\leq18 \\ 9y\leq-3x+18 \\ y\leq-\frac{3x}{9}+\frac{18}{9} \\ y\leq-\frac{1}{3}x+2 \end{gathered}[/tex]
Then, it is represented by the line that intercepts the y-axis at y=2. As this inequality has the symbol "<=" then, the line is included in the solution set, it is a solid line, and the solution are the y-values lower or equal to the line, then it is represented by:
The solution set to the system, is the space where the solutions of both inequalities overlaps, and it is represented by the purple zone:
The answer is option D.
