Floor Ceiling Floor Ceiling Floor Ceiling 2

$\left\lfloor x+\dfrac{1}{2}\right\rfloor +\lceil x \rceil+\left\lfloor x-\dfrac{1}{2}\right\rfloor\ge \left\lceil x+\dfrac{1}{2}\right\rceil+\lfloor x \rfloor+\left\lceil x-\dfrac{1}{2}\right\rceil$ Given that $1\le x\le 1000$ is a real number, find the number of different $x$'s that satisfy the above inequality.

If you believe there are an infinite number of $x$'s that satisfy the inequality, then submit $-1$.