If $$\left( {x + \frac{1}{x}} \right)$$   : $$\left( {x - \frac{1}{x}} \right)$$  = 5 : 3 the value of x is/are ?

$$\frac{{\left( {x + \frac{1}{x}} \right)}}{{\left( {x - \frac{1}{x}} \right)}} = \frac{5}{3}$$
By Componendo and Dividendo
$$\eqalign{ & \Rightarrow \frac{{x + \frac{1}{x} + x - \frac{1}{x}}}{{x + \frac{1}{x} - x + \frac{1}{x}}} = \frac{{5 + 3}}{{5 - 3}} \cr & \Rightarrow \frac{{2x}}{{\frac{2}{x}}} = \frac{8}{2} \cr & \Rightarrow {x^2} = 4 \cr & \Rightarrow x = \pm 2 \cr} $$