Mathmatical notation for maximum value in inequality

Question

I have an inequality using a chained notation which looks something like this:

$e \leq xa + b < f$, where $a,b,e,f,x \in \mathbb{N}$

What is the notation for a function, which returns the maximum value of $x$ for which the inequality is satisfied or zero if no such $x$ exists?

I was thinking of the following, but as I'm a newbie, I don't really know:

$g(a,b,e,f) = \begin{cases} x, & \text{if } \max_x{e \leq xa + b < f} \\0, & \text{otherwise} \end{cases}$

Thanks in advance!

Answer 1

I think $\max\left( \sup\left\lbrace x \in\mathbb{N}\vert e\leq xa+b\leq f\right\rbrace, 0\right)$ would work, since the supremum of the empty set is $-\infty$.