SOLUTION: Here is another subtracting rational expressions problem that is giving me some trouble. I am fine until I have factored and try to combine numerators. The minus sign throws me off

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Question 568499: Here is another subtracting rational expressions problem that is giving me some trouble. I am fine until I have factored and try to combine numerators. The minus sign throws me off. I am not sure what I am doing wrong, but I think it might be the order in which I am doing things. Do I multiply each numerator separately before combining, or maybe I'm switching the positives and negatives wrong. I need to know the correct order in which combine the numerators. One of my problems is:
3x+1/x-1-x-1/x-3+x+1/x^2-4x+3=
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
I assume your problem was:
(3x+1)/(x-1)-(x-1)/(x-3)+(x+1)/(x^2-4x+3)=%283x%2B1%29%2F%28x-1%29-+%28x-1%29+%2F%28x-3%29%2B%28x%2B1%29%2F%28%28x-1%29%28x-3%29%29 The long horizontal fraction lines come with invisible brackets enclosing the expressions in the numerator and the denominator. If you can only use slanted lines, and you fail to write all those brackets, it means something different. Then, math geeks misunderstand you and so do most calculators. I've seen people report wrong results at work because they forgot to type those brackets into the calculator.
Minus signs give everyone trouble, but I have tricks that help me deal with them. It is an unconventional way of working, and I found it is safer for me not to show it openly too often, but I’ll let you in on my secret. Minus signs are dangerous and should not be allowed to wander around unescorted. If we could keep them trapped within brackets, it would be better. I see your minus sign as part of an invisible (-1):
%283x%2B1%29%2F%28x-1%29-+%28x-1%29+%2F%28x-3%29%2B%28x%2B1%29%2F%28%28x-1%29%28x-3%29%29==
=
=
%283x%5E2-9x%2Bx-3-x%5E2%2B2x-1%2Bx%2B1%29%2F%28%28x-1%29%28x-3%29%29=

I do not usually write out the meaning of the minus signs so explicitly (too much work and too controversial), but I make an exception here for your benefit. I wrote some of those minus signs the way I see them in my mind. I believe that minus signs belong to the number that follows them, which may not always be visible to everyone, like that (-1) I am making visible for you. And of course I believe that there is no such thing as subtraction, it’s just adding a negative number. Unfortunately we are all indoctrinated to believe in subtraction since we are very young.