If $$\frac{{5x}}{{2{x^2} + 5x + 1}} = \frac{1}{3},$$     then the value of $$\left( {x + \frac{1}{{2x}}} \right) = \,?$$

$$\eqalign{ & \frac{{5x}}{{2{x^2} + 5x + 1}} = \frac{1}{3} \cr & \Rightarrow \frac{5}{{\frac{{2{x^2}}}{x} + \frac{{5x}}{x} + \frac{1}{x}}} = \frac{1}{3} \cr & \Rightarrow \frac{5}{{2x + \frac{1}{x} + 5}} = \frac{1}{3} \cr & \Rightarrow 2x + \frac{1}{x} + 5 = 15 \cr & \Rightarrow 2x + \frac{1}{x} = 10 \cr & {\text{Divide by 2 both sides }} \cr & \Rightarrow x + \frac{1}{{2x}} = \frac{{10}}{2} \cr & \Rightarrow 2x + \frac{1}{x} = 5 \cr} $$