SOLUTION: When it comes to solving Rational Functions, I get lost when I have a Sum and Product Problem on both sides of the equals sign, or when I get have to multiply to get numbers like x

Question 449838: When it comes to solving Rational Functions, I get lost when I have a Sum and Product Problem on both sides of the equals sign, or when I get have to multiply to get numbers like x^2. On my worksheet for homework, this problem is specifically giving me trouble.
(x/x-3)+(x-1/x^2-9)=(2x-3/x+3)
I've tried multiplying both sides by (x+3)(x-3), but i screw up everytime. Thanks for your help.
Answer by rwm(914) (Show Source): You can put this solution on YOUR website!
I assume the problem is really
(x)/(x-3)+(x-1)/(x^2-9)=(2x-3)/(x+3)
You don't want to multiply by (x+3)(x-3) which is (x^2-9)
You want to get the same denominator of (x^2-9)
so sometimes you will multiply by (x+3)/(x+3) which is 1/1
and sometimes by (x-3)/(x-3) which is also 1/1
We must discount any solutions where x=3 or -3
since they would lead to a zero denominator.
it ends up
(x^2+4 x-1)/((x-3) (x+3)) = (2 x-3)/(x+3)
((x-13) x+10)/(x^2-9) = 0
4/129 (x-13/2)^2 = 1
x = 1/2 (13-sqrt(129))
x = 1/2 (13+sqrt(129))
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