smc

If $x\le 2$, then $\lfloor x\rfloor +\lceil x\rceil \le 2+2<5$, so there are no solutions with $x\le 2$. If $x\ge 3$, then $\lfloor x\rfloor +\lceil x\rceil \ge 3+3$, so there are also no solutions here. Finally, if $2<x<3$, then $\lfloor x\rfloor +\lceil x\rceil =2+3=5$, so the solution set is $\boxed{\{x|2<x<3\}}$.