If \(n\) is an integer such that \(\frac{|n+6|-|n-3|}{\sqrt{100-n^3}} \geq 0\), then the number of possible values of \(n\) is __________

There are $6$ possible integers. The denominator must be positive, so $n^3 < 100$ gives $n\le4$ over the integers. The numerator is nonnegative when $|n+6|\ge|n-3|$, which after squaring gives $18n+27\ge0$ and hence $n\ge-1$ over the integers. There are six integers from $-1$ through $4$, and rounding the lower bound down to $-2$ would incorrectly give seven.