In the rectangular coordinate plane, the shaded region between the lines $y = 2 + 3x$ and $y = 12 - x$ and the positive $x$ and $y$-axes is given below.
Rectangles are constructed in this region with one side on the $x$-axis and one vertex each on the lines $y = 2 + 3x$ and $y = 12 - x$.
What is the length of the side on the $x$-axis of the rectangle with the largest area?
A) 6
B) $\dfrac{19}{3}$
C) $\dfrac{20}{3}$
D) 7
E) $\dfrac{22}{3}$