Let's simplify the given expression step-by-step:
Given expression:
[tex]\[ \left(\frac{x}{x-1} - \frac{1}{x+1}\right) \div \frac{x-1}{x^2-1} \][/tex]
First, let's work on the expression inside the parentheses:
[tex]\[ \frac{x}{x-1} - \frac{1}{x+1} \][/tex]
To combine these fractions, we need a common denominator. The common denominator for [tex]\( (x-1) \)[/tex] and [tex]\( (x+1) \)[/tex] is [tex]\( (x-1)(x+1) \)[/tex], which can also be written as [tex]\( x^2-1 \)[/tex].
Rewrite each fraction:
[tex]\[ \frac{x(x+1)}{(x-1)(x+1)} - \frac{(x-1)}{(x-1)(x+1)} \][/tex]
Simplify the numerators:
[tex]\[ \frac{x^2 + x}{x^2 - 1} - \frac{x - 1}{x^2 - 1} \][/tex]
Since the denominators are the same, subtract the numerators:
[tex]\[ \frac{(x^2 + x) - (x - 1)}{x^2 - 1} = \frac{x^2 + x - x + 1}{x^2 - 1} = \frac{x^2 + 1}{x^2 - 1} \][/tex]
Now, let's use this in the original division expression:
[tex]\[ \frac{x^2 + 1}{x^2 - 1} \div \frac{x-1}{x^2-1} \][/tex]
Dividing by a fraction is the same as multiplying by its reciprocal. So we have:
[tex]\[ \frac{x^2 + 1}{x^2 - 1} \times \frac{x^2 - 1}{x - 1} \][/tex]
The [tex]\( x^2 - 1 \)[/tex] terms cancel out:
[tex]\[ \frac{x^2 + 1}{x - 1} \][/tex]
Thus, the simplified expression is:
[tex]\[ \frac{x^2 + 1}{x - 1} \][/tex]
