Problem

The key is recognizing that $[x]$ is the floor (greatest integer ≤ $x$) and $(x)$ is the ceiling (least integer ≥ $x$), which differ only when $x$ is not an integer. If $x$ is an integer, both equal $x$ and the equation becomes trivial; if $x$ is not an integer, write $x = n + f$ where $n = [x]$ and $0 < f < 1$, then $(x) = n + 1$. Substituting into the equation gives you a constraint on $f$ that either yields a solution or forces $x$ to be an integer after all.

The key is recognizing that $[x]$ is the floor (greatest integer ≤ $x$) and $(x)$ is the ceiling (least integer ≥ $x$), which differ only when $x$ is not an integer. If $x$ is an integer, both equal $x$ and the equation becomes trivial; if $x$ is not an integer, write $x = n + f$ where $n = [x]$ and $0 < f < 1$, then $(x) = n + 1$. Substituting into the equation gives you a constraint on $f$ that either yields a solution or forces $x$ to be an integer after all.