jmc

We must have $$\lfloor x\rfloor -\{x\}=x-\lfloor x\rfloor ,$$or, simplifying the right-hand side, $$\lfloor x\rfloor -\{x\}=\{x\}.$$Thus, $$\lfloor x\rfloor =2\{x\}.$$Since the left-hand side is an integer, $2\{x\}$ must be an integer. We know that $0\le \{x\}<1,$ so either $\{x\}=0$ or $\{x\}=\frac{1}{2}.$ If $\{x\}=0,$ then $\lfloor x\rfloor =2\cdot 0=0,$ so $x=0,$ which is impossible because we are given that $x$ is nonzero. So we must have $\{x\}=\frac{1}{2},$ so $\lfloor x\rfloor =2\cdot \frac{1}{2}=1,$ and $x=1+\frac{1}{2}=\boxed{\frac{3}{2}}.$