perform the indicated operations and simplify x-1/x-9-x+1/x+9+x-153/x^2-81
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~ :)
To perform the indicated operations and simplify the expression, let's break it down step by step.
Given expression: (x - 1)/(x - 9) - (x + 1)/(x + 9) + (x - 153)/(x^2 - 81)
Step 1: Simplify the denominators.
The denominator of the first fraction (x - 9) and the denominator of the second fraction (x + 9) have a difference of squares pattern: (x^2 - 9^2). This can be factored as (x + 9)(x - 9).
The third fraction's denominator can also be factored as a difference of squares: (x^2 - 9^2) = (x + 9)(x - 9).
So the expression becomes: (x - 1)/(x + 9)(x - 9) - (x + 1)/(x + 9) + (x - 153)/(x + 9)(x - 9)
Step 2: Find a common denominator.
To add or subtract fractions, we need a common denominator. In this case, the common denominator is (x + 9)(x - 9).
Rewrite the expression with the common denominator: [(x - 1) - (x + 1)(x - 9) + (x - 153)] / (x + 9)(x - 9)
Step 3: Simplify and combine like terms.
Now let's simplify and combine like terms.
[(x - 1) - (x^2 - 10x + 9) + (x - 153)] / (x + 9)(x - 9)
In the second term, distribute the negative sign to each term within the parentheses:
(x - 1 - x^2 + 10x - 9 + x - 153) / (x + 9)(x - 9)
Combine like terms:
(-x^2 + 11x - 17) / (x + 9)(x - 9)
And that's the simplified expression:
(-x^2 + 11x - 17) / (x + 9)(x - 9)