Writing the sum of two rational functions as a single rational function.

Question

Write as a single fraction:

$$\frac{2x}{x-1} - \frac{x}{x+1}$$

Please can somebody talk me through this question as I don't understand how to get a common denominator. Thank you.

Answer 1

The key thing to note is that $\cfrac {x+1}{x+1}=\cfrac {x-1}{x-1}=1$ so we can multiply by each of these forms without changing the value (a little care may be needed about the denominator being zero, but you can catch up with that at the end).

So rewrite as $$\frac {2x}{x-1}\cdot\frac {x+1}{x+1}-\frac x{x+1}\cdot \frac{x-1}{x-1}$$

Note that like normal rational fractions we've chosen the forms of fraction which make the denominators equal. Now simplify numerator and denominator.

Answer 2

it is $\frac{2x}{x-1}-\frac{x}{x+1}$? we get $\frac{2x(x+1)-x(x-1)}{(x-1)(x+1)}=\frac{x (x+3)}{(x-1) (x+1)}$