Q. Find domain: $\lfloor \dfrac3x \rfloor + \lfloor \dfrac4x \rfloor = 5$
My approach:
$\dfrac3x - 1 ≤ \lfloor \dfrac3x \rfloor ≤ \dfrac3x$
$\dfrac4x - 1 ≤ \lfloor \dfrac4x \rfloor ≤ \dfrac4x$
Adding them,
$\dfrac7x - 2 ≤ \lfloor \dfrac3x \rfloor + \lfloor \dfrac4x \rfloor ≤ \dfrac7x$
$\dfrac7x - 2 ≤ 5 ≤ \dfrac7x$
$x \in (1, \frac75]$