Why does $-\frac{1}{17-x}$ equal $\frac{1}{x-17}$? 
Why does $-\frac{1}{17-x}$ equal $\frac{1}{x-17}$?

Is there any simple computation to make this seem a little bit more intuitive? Right now, I cannot wrap my head around the fact that I can just switch signs of the term in the denominator.
 A: It comes from the following algebra facts:
$$-\frac{a}{b}=\frac{-a}{b}=\frac{a}{-b}$$
and
$$\begin{align}
-(c-d) &= (-c)-(-d)\\
&=-c+d\\
&=d-c
\end{align}$$
(For this second fact we first "distribute the negative sign", and then "minus a negative is plus".)
Are you familiar with those manipulations? The person who wrote the text you got this from just combined a few steps into one equation.
Generally, unless you're first being introduced to such algebra, it's not considered necessary to write down every step - it's like they're saying "well, we start over here, and end up over there", and expect that you can fill in the details. And after a few times where you write it out explicitly, you'll find you can "do it with just your eyes", as you are reading along.

EDIT: (Since this is currently the accepted answer, I'll add another pattern that also sometimes shows up, and can be briefly puzzling, for the benefit of future readers.)
Similarly
$$\frac{a}{b} = \frac{-a}{-b}$$
is pretty obvious written that way, but can be harder to recognize when it's used to rewrite
$$\frac{1-x}{1-2x}$$
as $$\frac{x-1}{2x-1}$$
A: $17-x=-1(x-17) $ so $(-1) (17-x) =(-1) ^2(x-17) =x-17$
A: You can break this down into a couple of basic facts. First of all, we have $\frac{1}{-1}=-1$. Thus, if you want to see the opposite, or negative, of a fraction, we can multiply either the numerator or denominator by $-1$.
Secondly, and this might be the part you're really asking about, $a-b$ and $b-a$ are opposites. Observe: $$-1(a-b) = -a - (-b) = -a+b = b-a$$
Even better, think of numbers: What's $5-3$? What's $3-5$? To do the second computation, don't you just do the first one, and then include a minus sign, since the order was "wrong" for doing subtraction as an intuitive "take-away"?
Putting these together: to get the negative of a fraction, you can negate either the numerator or the denominator, and we choose the denominator. To negate a difference, you can just flip the subtraction statement around.
Does this help?
