I am working on modeling some data, and I've obtained an expression that is the ratio of two upper incomplete gamma functions:
$$p = \frac{\Gamma(a,x_\text{signal})}{\Gamma(a,x_\text{noise})}$$
The parameterization of the gamma function is the same as on Wolfram|Alpha: https://mathworld.wolfram.com/IncompleteGammaFunction.html. I would like to rearrange this equation to solve for the value of $x_{signal}$. I suspect this is not possible, but I do not have the mathematical skills to be certain. I've tried converting both upper incomplete gamma functions into their equivalent integral representations, but that doesn't provide me with any obvious way forward:
$$p = \frac{\int_{t=x_\text{signal}}^\infty t^{a-1}\cdot e^{-t} dt}{\int_{t=x_\text{noise}}^\infty t^{a-1}\cdot e^{-t} dt}$$
Any help (including confirmation that such a solution is not obtainable) would be greatly appreciated.
EDIT:
In case it helps, $p = 0.4390$ in my case. The value of $a$ is unknown, except that it is $\geq 1$. Also, both $x_{signal}$ and $x_{noise}$ are $\geq0$, with $x_{signal} \geq x_{noise}$.