closed-form expression for solution of inequality $ \left\lceil \frac{a}{x} \right\rceil > b$

Question

Considering the inequality where $a, b > 0$, $$ \left\lceil \frac{a}{x} \right\rceil > b, $$ is there a closed-form expression for $x$?

Answer 1

$\lceil y\rceil >b$ is equivalent to $y>\lfloor b\rfloor,$ so your inequality is equivalent to $\frac ax>\lfloor b\rfloor.$ Since $a,b>0,$ this means $x>0,$ and thus the answer is $0<x<\frac a{\lfloor b\rfloor}.$

The only problem case is when $\lfloor b\rfloor=0.$ Then any $x>0$ is a solution.