How many whole numbers $n$ have a difference $|\sqrt{n}-10| < 1$?

Question

Question

The difference of $\sqrt{n}$ and $10$ is less than $1$. How many whole numbers $n$ are there with this characteristic? $$(A)\ 15 \ \ \ \ \ \ (B)\ 29 \ \ \ \ \ \ (C)\ 39 \ \ \ \ \ \ (D)\ 24 \ \ \ \ \ \ (E)\ 26 $$

What I Am Stuck On:

First, know that whole numbers do not include negative or imaginary numbers, and zero is inside this category. I infer that this problem means that the difference could be negative, so the numbers that satisfy this condition are from $120$ to $0$ inclusive. So there are $120-0+1=121$ whole numbers. What went wrong?

Answer 1

We suppose $\sqrt n=r$. Then, $|r−10|<1$ is the mathematical formulation of above statement. We thus get $9<\sqrt n=r<11$. Since all three sides of this inequality deal with positive numbers; we can square. So

$$81<n<121$$

Positive integers satisfying it are $\{82,83,84,\ldots,120\}$.

That is $9+10+10+10=39$ positive integers.