Let $a,c>0$, $b>1$ be constants. I am wondering what kind of necessary conditions we can find in order that
$$ e^{ax} \geq b - cx$$
holds. So I would like something like
$$ e^{ax} \geq b - cx \Rightarrow x \geq f(a,b,c)$$
where $f$ is some fairly simple function (not something esoteric like the Lambert W-function).
Graphically speaking, it is clear that the equation
$$ e^{ax} = b - cx$$
has precisely one solution, which is positive.
Many thanks for your help.