This is the problem: "A mountain climber is at an elevation of $10,000$ feet. If she descends $2,000$ feet a day, which equation would be used to show how many days it will take to reach sea level ($0$ feet)?"
My online school says the problem looks like this; $10,000 ÷ 2,000$ but I thought it should be $-2,000$ since climber is descending.
The online teacher said this, "It would not be a negative number since we are given $10,000$ feet and $2,000$ each day. This answer is correct."
On
If you define "going down" as negative, then the mountain climber is trying to travel $10 \ 000$ feet down, which is $-10 \ 000$. If she's going down $-2 \ 000$ a day, then you have:
$$\text{days} = \dfrac{-10 \ 000}{- 2 \ 000}$$
On the other hand, if you define "going down" as positive, then the mountain climber is still traveling $10 \ 000$ feet down, but it's defined as positive. If going down is positive, so is going down at a rate of $2 \ 000$ a day:
$$\text{days} = \dfrac{10 \ 000}{ 2 \ 000}$$
Both your answer and the teacher's answer are correct, but yours feels more natural to me.
I just noticed: her velocity is not given as $2 \ 000$, that's her speed. Her velocity is a vector quantity and so is given by ${2 \ 000}\dfrac {\text{ft}}{\text{day}}\left[{\text{towards the ground}}\right]$, and you have to write that whichever way is useful in solving the problem. In these word problems, it's often a matter of preference which way is positive and which way is negative.
$\frac{10,000}{2,000}$ is defined to be the solution to $10,000-2,000x=0$.
Now it's left to prove that the solution to $10,000-2,000x=0$ should be the answer.
Well, after $1$ day we have $10,000-2,000=8,000$ feet left, which is intuitive -- if she descends $2,000$ a day, then after one day she will have $8,000$ left.
So intuitively $x$, i.e. the solution to $10,000-2,000x=0$, should indeed be the answer, but, as I said, it is by the definition of division equal to $\frac{10,000}{2,000}$.