Question

Find the difference of x+1/x-1 x-8/x-1

Answer 1

To find the difference of the given expression, we can first simplify each term separately and then subtract them.

First, we simplify x+1/x-1 and x-8/x-1.

For x+1/x-1:
(x+1)/(x-1) can be simplified by multiplying the numerator and the denominator by (x+1) to eliminate the fraction:
[(x+1)(x+1)]/(x-1)(x+1) = (x^2 + 2x + 1)/(x^2 - 1)

For x-8/x-1:
(x-8)/(x-1) is already simplified.

Now, we can find the difference between the two expressions:
(x^2 + 2x + 1)/(x^2 - 1) - (x-8)/(x-1)

To subtract these fractions, we need a common denominator:
The common denominator is (x^2 - 1)(x-1) since (x-1) is already a factor in the second fraction, and (x^2 - 1) includes all the necessary factors.

Rewriting the fractions with the common denominator:
[(x^2 + 2x + 1)(x-1)]/((x^2 - 1)(x-1)) - (x-8)(x^2 - 1)/((x-1)(x^2 - 1))

Expanding the numerator in the first fraction:
(x^3 - x^2 + 2x^2 - 2x + x - 1)/((x^2 - 1)(x-1)) - (x^3 + 8x^2 - x - 8)/((x-1)(x^2 - 1))

Combining like terms in each fraction:
(x^3 + x^2 - x - 1)/((x^2 - 1)(x-1)) - (x^3 + 8x^2 - x - 8)/((x-1)(x^2 - 1))

Now, we can subtract the fractions:
((x^3 + x^2 - x - 1) - (x^3 + 8x^2 - x - 8))/((x-1)(x^2 - 1))

Simplifying the numerator:
(x^3 + x^2 - x - 1 - x^3 - 8x^2 + x + 8)/((x-1)(x^2 - 1))

Combining like terms in the numerator:
(-7x^2 + 7)/((x-1)(x^2 - 1))

Factoring out -7 from the numerator:
7(-x^2 + 1)/((x-1)(x^2 - 1))

The final difference is 7(-x^2 + 1)/((x-1)(x^2 - 1))