Combining rationals and creating a function

Mar 2017
356
3
Massachusetts
Hi,

I hope someone can help. I want to know whether there is any rationale behind my textbooks solution for combining rationals to create a function. In all the solutions in question 3 (which I have attached), the rational on the right-side is always brought to the left-side. For example, 3d's textbook solution is f(x) = ((x-2)/(x+3)) - ((x-4)/(x+5)). What I'm wondering is would the solution be just as correct if I were to subtract the left-side from the right-side? Thereby making 3d's solution f(x) = ((x-4)/(x+5)) - ((x-2)/(x+3)) instead of the textbook solution mentioned.

Screen Shot 2017-05-03 at 7.14.22 PM.png

I know that have the same zeros, but my textbook seems to only consider the solution where the right-side is brought of to the left.

Any ideas on why this is the case? Or does it not matter?

Sincerely,
Olivia
 

skeeter

MHF Helper
Jun 2008
16,217
6,765
North Texas
What I'm wondering is would the solution be just as correct if I were to subtract the left-side from the right-side? Thereby making 3d's solution f(x) = ((x-4)/(x+5)) - ((x-2)/(x+3)) instead of the textbook solution mentioned.
The two functions would be opposite in sign, but they would have the same zeros and asymptotes.
 

Attachments

SlipEternal

MHF Helper
Nov 2010
3,728
1,571
I misunderstood the question. Yes, you would still have found a solution. There is an infinite family of solutions to each of those problems.