Using properties of proportion, solve for x :

$\dfrac{\sqrt{x + 5} + \sqrt{x - 16}}{\sqrt{x + 5} - \sqrt{x - 16}} = \dfrac{7}{3}.$

Given,

$\Rightarrow \dfrac{\sqrt{x + 5} + \sqrt{x - 16}}{\sqrt{x + 5} - \sqrt{x - 16}} = \dfrac{7}{3}$

Applying componendo and dividendo:

$\Rightarrow \dfrac{\sqrt{x + 5} + \sqrt{x - 16} + \sqrt{x + 5} - \sqrt{x - 16}}{\sqrt{x + 5} + \sqrt{x - 16} - (\sqrt{x + 5} - \sqrt{x - 16})} = \dfrac{7 + 3}{7 - 3} \\[1em] \Rightarrow \dfrac{2\sqrt{x + 5}}{2\sqrt{x - 16}} = \dfrac{10}{4} \\[1em] \Rightarrow \dfrac{\sqrt{x + 5}}{\sqrt{x - 16}} = \dfrac{10}{4} \\[1em] \Rightarrow \dfrac{x + 5}{x - 16} = \dfrac{100}{16} \\[1em] \Rightarrow 16(x + 5) = 100(x - 16) \\[1em] \Rightarrow 16x + 80 = 100x - 1600 \\[1em] \Rightarrow 100x - 16x = 1680 \\[1em] \Rightarrow 84x = 1680 \\[1em] \Rightarrow x = 20.$

Hence, x = 20.