Is this equation solvable (by hand) and what do you call this equation?
$a = x - \min(b\, x, c)$
$a, b, c$ are known. We need to solve for $x$.
If it is, and I can get guidance as to what this is called, and where there are resources to find out how to solve, this, I'm happy to do this.
Expanding on the comment by player100:
The function $f(x) = x-\min(bx,c) $ has two regions:
$f(x) = (1-b)x \quad \text{if } bx \le c$
$f(x) = x-c \quad \text{if } bx \gt c$
Setting $f(x) = a$ then we have
$x = \frac{a}{1-b} \quad \text{if } \frac{ab}{1-b} \le c$
$x = c+a \quad \text{if } b(c+a) \gt c$
Note that we can re-arrange the condition $b(c+a) \gt b$ as $ab \gt (1-b)c$, so the two conditions are mutually exclusive if $1-b > 0$ i.e. $b < 1$. So if $b< 1$ then the equation $f(x)=a$ has exactly one solution.
However, if $b > 1$ then the equation $f(x)=a$ will have two solutions if $ab > (1-b)c$, one solution if $ab = (1-b)c$ and no solutions if $ab < (1-b)c$.