Final answer:
To simplify (7x)/(x+1)+(7)/(x²-1), factor the second term's denominator to (x+1)(x-1) to get a common denominator for both fractions. Then combine and simplify the fractions, resulting in (7x² - 7x + 7)/((x+1)(x-1)), which is the fully simplified form.
Step-by-step explanation:
To fully simplify the expression (7x)/(x+1) + (7)/(x²-1), we need to recognize that x²-1 is a difference of squares and can be factored into (x+1)(x-1). This allows us to have a common denominator for both fractions. Consequently, the simplified expression should have the denominator (x+1)(x-1).
First, we factor the second term:
(7)/(x²-1) = (7)/((x+1)(x-1)).
Then, to combine the fractions, we rewrite the first fraction to have the same denominator:
(7x)/((x+1)(1)) * ((x-1)/(x-1)) = (7x(x-1))/((x+1)(x-1)).
Now, we add the fractions with a common denominator:
((7x(x-1)) + 7)/((x+1)(x-1)).
Expanding the numerator, we get:
(7x² - 7x + 7)/((x+1)(x-1)).
After expanding, there are no like terms to combine or eliminate, therefore, this is the fully simplified form of the expression. Checking the answer, we see that it is reasonable as it combines both original fractions into a single fraction with a common denominator.