what are the rules/rationale for "simplifying" negative denominators

Question

So I'm losing my mind trying to understand the rules and rationale behind "simplifying" expressions and equations in Algebra. It's been decades since I've had to study for it (going back and rehashing it out at Khan Academy) and I just don't understand what the rules are for determining whether or not an equation is "simplified." Take this one for example:

      -2b - 11
a =  -----------
     8b - 5c + 9

Now according to Khan Academy and this practice question, the simplified version of this problem would be:

      2b + 11
a =  -----------
     -8b + 5c - 9

Why is this considered simplified? All I see is a few negatives flipped around and cannot imagine for the life of me what makes one more simple than the other?

Answer 1

The most likely reason why that is the preferred answer is that in the first equation, the numerator has two negatives, so it's easier to read if you simply change both of them to positives.

Another reason might be that when you're trying to "simplify" a polynomial or an expression like this you want to have the top and bottom start with a positive term, and this is accomplished in the second solution (notice the denominator could be rearranged to start with a positive term).

In short, having all terms negative in a polynomial is like bad math grammar, when you can just simply multiply everything by -1 and make it look nice and pleasant to the eye.

Answer 2

Nothing. Both are equally simplified.

Edit: Take Khan Academy problems with a grain of salt. Especially anything graded by a computer.

Answer 3

These expressions are equal. Which is "simpler" is in the eye of the beholder. The second has one fewer minus sign-maybe that is what they are looking at. You might even rewrite the denominator as $5c-8b-9$ to get rid of the leading negative sign. A matter of taste.