### Instruction: How many integer values of $n$ satisfy $-50 < n^3 < 50$?### Response: I want to find the possible values of $n$ that make $n^3$ lie between $-50$ and $50$.
I could try to use some algebra, but it might be easier to just try some values of $n$ and see what happens.
I know that $n^3$ changes sign when $n$ changes sign, so I can start by looking at positive values of $n$.
If $n = 1$, then $n^3 = 1$, which is in the range.
If $n = 2$, then $n^3 = 8$, which is still in the range.
If $n = 3$, then $n^3 = 27$, which is also in the range.
If $n = 4$, then $n^3 = 64$, which is too big.
So the positive values of $n$ that work are $n = 1, 2, 3$.
For negative values of $n$, I can use the fact that $(-n)^3 = -n^3$.
So if $n = -1$, then $n^3 = -1$, which is in the range.
If $n = -2$, then $n^3 = -8$, which is still in the range.
If $n = -3$, then $n^3 = -27$, which is also in the range.
If $n = -4$, then $n^3 = -64$, which is too small.
So the negative values of $n$ that work are $n = -1, -2, -3$.
Finally, I have to consider the case where $n = 0$, which also makes $n^3 = 0$, which is in the range.
So the total number of integer values of $n$ that satisfy the inequality is $7$.