To solve this equation, we first need to find a common denominator for the fractions on the left side of the equation.

First fraction:
1 / (x^2 - 1)

Second fraction:
2 / (x^2 - 1)

So, the common denominator is (x^2 - 1).

Now, we rewrite the fractions with the common denominator:

(1) / (x^2 - 1) - (2) / (x^2 - 1) = 1 / (x - 1)

Now, combine the fractions:

(1 - 2) / (x^2 - 1) = 1 / (x - 1)

-1 / (x^2 - 1) = 1 / (x - 1)

Now, cross multiply:

Expand both sides:

Rearrange the equation:

x^2 + x - 2 = 0

Now, solve the quadratic equation for x:

x = (-1 ± √(1 + 8)) / 2

x = (-1 ± √9) / 2

x = (-1 ± 3) / 2

x = -4 / 2 = -2 or x = 2 / 2 = 1

Therefore, the values of x that satisfy the equation are x = -2 and x = 1.