You may want to review the case with a common denominator and finding common denominators first.
If the two rational functions you want to add or subtract don't have a common denominator, you need to rewrite them so they do, just as with fractions. For example, if you have to subtract (x + 1)/(x − 1) − x, the common denominator is x − 1. We need to rewrite x so it has this denominator too, so we write it as x(x − 1)/(x − 1) = (x2 − x)/(x − 1). Now subtracting gives us (−x2 + 2x + 1)/(x − 1). Finally, we look for common factors of the numerator and denominator to reduce; there aren't any, so we're done.