SOLUTION: ((x-3)/(x^2-2x-15))+((x+2)/(5-x))

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Question 151424: ((x-3)/(x^2-2x-15))+((x+2)/(5-x))
Answer by jim_thompson5910(14933) About Me  (Show Source):
You can put this solution on YOUR website!
%28x-3%29%2F%28x%5E2-2x-15%29%2B%28x%2B2%29%2F%285-x%29 Start with the given expression


%28x-3%29%2F%28%28x%2B3%29%28x-5%29%29%2B%28x%2B2%29%2F%285-x%29 Factor x%5E2-2x-15 to get %28x%2B3%29%28x-5%29


%28x-3%29%2F%28%28x%2B3%29%28x-5%29%29%2B%28x%2B2%29%2F%28-1%28x-5%29%29 Factor 5-x to get -1%28x-5%29


%28x-3%29%2F%28%28x%2B3%29%28x-5%29%29%2B%28-x-2%29%2F%28x-5%29 Simplify.


Now let's find the LCM of the denominators %28x%2B3%29%28x-5%29 and %28x-5%29. It turns out that the LCM of these denominators is %28x%2B3%29%28x-5%29. So the goal is to get both denominators to the LCD %28x%2B3%29%28x-5%29


%28x-3%29%2F%28%28x%2B3%29%28x-5%29%29%2B%28%28x%2B3%29%2F%28x%2B3%29%29%28%28-x-2%29%2F%28%28x-5%29%29%29 Multiply the 2nd fraction by %28%28x%2B3%29%29%2F%28%28x%2B3%29%29


%28x-3%29%2F%28%28x%2B3%29%28x-5%29%29%2B%28%28x%2B3%29%28-x-2%29%29%2F%28%28x%2B3%29%28x-5%29%29%29 Combine the fractions.


%28x-3%29%2F%28%28x%2B3%29%28x-5%29%29%2B%28-x%5E2-2x-3x-6%29%2F%28%28x%2B3%29%28x-5%29%29%29 FOIL


%28x-3%29%2F%28%28x%2B3%29%28x-5%29%29%2B%28-x%5E2-5x-6%29%2F%28%28x%2B3%29%28x-5%29%29%29 Combine like terms.


%28x-3-x%5E2-5x-6%29%2F%28%28x%2B3%29%28x-5%29%29%29 Add the fractions.


%28-x%5E2-4x-9%29%2F%28%28x%2B3%29%28x-5%29%29%29 Combine like terms.


%28-x%5E2-4x-9%29%2F%28x%5E2-2x-15%29 FOIL the denominator.


So %28x-3%29%2F%28x%5E2-2x-15%29%2B%28x%2B2%29%2F%285-x%29 adds and simplifies to %28-x%5E2-4x-9%29%2F%28x%5E2-2x-15%29


In other words, %28x-3%29%2F%28x%5E2-2x-15%29%2B%28x%2B2%29%2F%285-x%29=%28-x%5E2-4x-9%29%2F%28x%5E2-2x-15%29 where x%3C%3E-3 or x%3C%3E5