\(\displaystyle \dfrac{f(x)-f(7)}{x - 7} = \dfrac{\frac{4}{x+1}-\frac{4}{7+1}}{x - 7} =\)
\(\displaystyle \dfrac{\frac{8*4}{8(x+1)}- \frac{4(x+1)}{8(x+1)}}{x - 7} =\)
\(\displaystyle \dfrac{\frac{32 - 4x - 4}{8(x+1)}}{x-7}} =\)
\(\displaystyle \dfrac{\frac{28-4x}{8(x+1)}}{x-7} = \dfrac{\frac{3.5 -.5x}{x+1}}{x-7}\) So you have calculated correctly.
Unfortunately, though correct, it is not the most promising line of attack. Try this instead
\(\displaystyle \dfrac{\frac{28x -4x}{8(x+1)}}{x-7} = \dfrac{\frac{-4(x-7)}{8(x+1)}}{x-7} = \dfrac{\frac{x-7}{2(x+1)}}{x-7}\)
You probably can solve this by sight. But, in the interest of completeness, the formal way to proceed is to take advantage of two facts
\(\displaystyle m = \dfrac{m}{1}\) and \(\displaystyle \dfrac{\frac{a}{b}}{\frac{c}{d}} = \dfrac{a}{b} * \dfrac{d}{c} = \dfrac{ad}{bc}\)
See what you get
