Combining rationals and creating a function

otownsend

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.

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

SlipEternal

MHF Helper
Is 5-3 the same as 3-5?

skeeter

MHF Helper
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.

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SlipEternal

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