If $\Phi$ is the CDF of the standard normal distribution, then I have the following equation, which I'm trying to solve for $x$:
$a\Phi(x)=1-\Phi(x+b)$,
where $a$ and $b$ are real, positive constants.
I tried writing out the actual integral over $\varphi$, of the form $\int_{-\infty}^{x}e^{-(t-b)^2}dt$ but then you end up with a term of the for $e^{2bt}$ inside the integral.
Is the above equation actually solvable? (E.g. under the special case of $b=0$, the solution becomes $x=\Phi^{-1}\left( \frac{1}{a+1} \right)$.) If it is: how? If it isn't: why not?