How to solve ( (1/x) - 1 )/(x-1)?

Question

Simplify

( (1/x) - 1 ) / (x - 1) .

Answer 1

Carry out the subtraction in the numerator first:

(1/x) - 1 = (1/x) - (x/x) = (1-x)/x

Now you have

[(1-x)/x]/(x - 1) = [(1 - x)/x] * 1/(x - 1) = (1 - x)/[x(x - 1)]

You might notice that (1 - x) = - (x - 1).  If you're not sure of that, convince yourself.  Try plugging in numbers.  In any event, the 1 - x cancels the x - 1, leaving -1, so the whole fraction reduces to

- 1/x