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 \(-1\).

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