SOLUTION: i need to solve for this equation: ((x+1)/x-2))+((x-3)/(x-1))<0

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Question 588367: i need to solve for this equation:
((x+1)/x-2))+((x-3)/(x-1))<0
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
%28x%2B1%29%2F%28x-2%29%2B%28x-3%29%2F%28x-1%29%3C0
Maybe we could get a common denominator:

And we could do those multiplications in the numerator:
%28x%5E2-1%29%2F%28%28x-2%29%28x-1%29%29%2B%28x%5E2-5x%2B6%29%2F%28%28x-2%29%28x-1%29%29%3C0
And we could add the two fractions:
%282x%5E2-5x%2B5%29%2F%28%28x-2%29%28x-1%29%29%3C0
The numerator is always positive.
We can see that 2x%5E2-5x%2B5=0 has no real roots because the discriminant is negative:
%28-5%29%5E2-4%2A2%2A5=25-40=-15%3C0
2x%5E2-5x%2B5 is positive for x=0 (it's 5), and everywhere else.
For %282x%5E2-5x%2B5%29%2F%28%28x-2%29%28x-1%29%29 to be negative, the denominator has to be negative.
Each factor has a zero (x=1 and x=2), is negative for lesser x values and positive for greater values.
The product will be negative for x such that 1%3Cx%3C2.
For x=1 and x=2, the denominator is zero and the functions do not exist.
For x<1 and x>2 the function is positive.
So the solution is highlight%281%3Cx%3C2%29