How to solve this difficult one variable equation analytically?

Question

Would anybody like to explain me clearly how to solve analytically this equation?

$$5.56=\frac{1-e^{-5.5x}}{1-e^{-x}}$$

I have already solved it with Mathematica and it gives $x=-0.004809$. However, I would like to know the methods for solving it analytically as a good mathematician does.

Cheers, Endora

Answer 1

Let $y=e^{-0.5x}$, then

\begin{align*} 5.56 &= \frac{1-y^{11}}{1-y^{2}} \\ 5.56 (1+y) &= 1+y+y^{2}+\ldots+y^{10} \\ \end{align*}

$y$ is unsolvable in terms of radicals, though you can solve it numerically.