Simplify:
(x)/(x-1)-(1)/(x+1)
To simplify the expression (x)/(x-1) - (1)/(x+1), we first need to find a common denominator for both fractions. The common denominator for (x-1) and (x+1) is (x-1)(x+1).
Now, we'll convert each fraction to have the common denominator:
(x)/(x-1) = (x)(x+1)/[(x-1)(x+1)]
(1)/(x+1) = (1)(x-1)/[(x-1)(x+1)]
With the fractions having a common denominator, we can now combine them:
[(x)(x+1)]/[(x-1)(x+1)] - [(1)(x-1)]/[(x-1)(x+1)]
Next, we can subtract the fractions. When subtracting fractions, we subtract the numerators (top numbers) and keep the common denominator:
[(x)(x+1) - (1)(x-1)]/[(x-1)(x+1)]
Expanding the expressions in the numerator:
[x^2 + x - (x-1)]/[(x-1)(x+1)]
Simplifying further by combining like terms:
[x^2 + x - x + 1]/[(x-1)(x+1)]
This simplifies to:
[x^2 + 1]/[(x-1)(x+1)]
So the simplified expression is (x^2 + 1)/[(x-1)(x+1)].