$(\begin{array}{l} \frac{{\sqrt {3{\rm{x\;}} + {\rm{\;}}4} {\rm{\;}} + {\rm{\;}}\sqrt {3{\rm{x}}\; - \;5} }}{{\sqrt {3{\rm{x\;}}+ {\rm{\;}}4{\rm{\;}}}\; - \;\sqrt {3{\rm{x}}\; - \;5} }} = 9\\ \Rightarrow {\rm{\;}}\frac{{\sqrt {3{\rm{x\;}} + {\rm{\;}}4} {\rm{\;}} + {\rm{\;}}\sqrt {3{\rm{x}} \;-\; 5} }}{{\sqrt {3{\rm{x\;}} + {\rm{\;}}4{\rm{\;}}} \;-\; \sqrt {3{\rm{x}}\; - \;5} }}\; = \;\frac{9}{1} \end{array})$

using componendo and dividendo, we get

$(\frac{{(\sqrt {3{\rm{x\;}} + {\rm{\;}}4} {\rm{\;}} + {\rm{\;}}\sqrt {3{\rm{x}} - 5)} {\rm{\;}} + {\rm{\;}}(\sqrt {3{\rm{x\;}} + {\rm{\;}}4} - \sqrt {3{\rm{x}} - 5)} }}{{(\sqrt {3{\rm{x\;}} + {\rm{\;}}4{\rm{\;}}} {\rm{\;}} + {\rm{\;}}\sqrt {3{\rm{x}} - 5)} - (\sqrt {3{\rm{x\;}} + {\rm{\;}}4} - \sqrt {3{\rm{x}} - 5)} }} = \frac{{9{\rm{\;}} + {\rm{\;}}1}}{{9 - 1}}{\rm{\;}})$

$(\Rightarrow {\rm{\;}}\frac{{2\sqrt {3{\rm{x\;}} + {\rm{\;}}4} }}{{2\sqrt {3{\rm{x}} - 5} }} = \frac{{10}}{8} \Rightarrow \;\frac{{\sqrt {3{\rm{x\;}} + {\rm{\;}}4} }}{{\sqrt {3{\rm{x}} - 5} }} = \frac{5}{4})$

$(\Rightarrow {\rm{\;}}\frac{{3{\rm{x\;}} + {\rm{\;}}4}}{{3{\rm{x}} - 5}} = \frac{{25}}{{16}})$ (On squaring both sides)

⇒ 75x – 125 = 48x + 64

⇒ 75x – 48x = 64 + 125

⇒ 27x = 189 ⇒ x = 7.