Question

perform the indicated operations and simplify x-1/x-9-x+1/x+9+x-153/x^2-81

Answer 1

let's put parenthesis so that it's not so confusing:
(x-1)/(x-9) - (x+1)/(x+9) + (x-153)/(x^2-81)
first we get the least common denominator (LCD) of these,, observe that (x-9)*(x+9) = x^2 - 81 . thus the LCD is equal to x^2 - 81 . multiplying all terms by (x^2-81)/(x^2-81) or (x-9)(x+9)/(x-9)(x+9) , some terms will be cancelled:
(x-1)(x+9)/(x^2-81) - (x+1)(x-9)/(x^2-81) + (x-153)/(x^2-81)
simplifying, and combining the numerator terms:
[ (x-1)(x+9) - (x+1)(x-9) + x-153 ]/(x^2-81)
[ (x^2 + 8x - 9 - x^2 + 8x + 9 + x - 153 ]/(x^2 - 81)
note that x^2 and -x^2 and also the 9 and -9 will be cancelled:
(16x + x - 153)/(x^2-81)
(17x - 153)/(x^2-81)
we can factor out 17 from the numerator:
17(x-9)/(x^2-81)
17(x-9)/[(x-9)(x+9)]
17/(x+9)

hope this helps~ :)