Question

How do you add or subtract `1/(x²-x) - 1/x`?

Answer 1

The given expression simplifies to `(-x + 2)/(x^2-x)`

Explanation:

I will let the given expression be represented by `P`. So that

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

You need to manipulate (following the rules of algebra) the terms
`1/(x^2 - x) and 1/x`
so that they have the same denominator.

Multiply top and bottom of `1/(x^2 - x)` by x. That yields

`x/(x*(x^2-x))`

Multiply top and bottom of `1/x` by (x^2 - x). That yields

`(x^2 - x)/(x*(x^2-x))`

Putting them together,

`P = x/(x*(x^2-x)) - (x^2 - x)/(x*(x^2-x))`

A few more simplification steps to go

`P = (x - (x^2 - x))/(x*(x^2-x)) = (-x^2 + 2*x)/(x*(x^2-x))`

`P = (x * (-x + 2))/(x*(x^2-x)) = (cancel(x) * (-x + 2))/(cancel(x)*(x^2-x))`

So `P = (-x + 2)/(x^2-x)` seems to be as far as simplification will take it.

I hope this helps,
Steve