Floor Ceiling Floor Ceiling Floor Ceiling

Algebra Level 5

$\left\lfloor x+\dfrac{1}{2}\right\rfloor \lceil x \rceil\left\lfloor x-\dfrac{1}{2}\right\rfloor\ge \left\lceil x+\dfrac{1}{2}\right\rceil\lfloor x \rfloor\left\lceil x-\dfrac{1}{2}\right\rceil$ 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 $-1$ .