jmc

Observe that $\lfloor 2x\rfloor$ is an integer, so it follows that $\lfloor \lfloor 2x\rfloor -1/2\rfloor =\lfloor 2x\rfloor -1$. Also, $\lfloor x+2\rfloor =\lfloor x\rfloor +2$. Thus, our equation becomes $$\lfloor 2x\rfloor =\lfloor x\rfloor +3.$$Let $n=\lfloor x\rfloor ,$ so $n\le x<n+1.$

If $x<n+\frac{1}{2},$ then $2n\le x<2n+1,$ so $\lfloor 2x\rfloor =2n,$ and $$2n=n+3,$$which means $n=3.$

If $x\ge n+\frac{1}{2},$ then $2n+1\le x<2n+2,$ so $\lfloor 2x\rfloor =2n+1,$ and $$2n+1=n+3,$$which means $n=2.$

Therefore, the set of solutions is $x\in \boxed{\left[\frac{5}{2},\frac{7}{2}\right)}.$