Hi,
I'm not seeing how these two are equivalent: $\displaystyle -\frac{1}{x-1} \equiv \frac{1}{1-x}$.
Could somebody show/tell me, please?
We have $\displaystyle a\cdot(-b)=(-a)\cdot b=-ab$. In particular, when $\displaystyle b = 1/c$, we have $\displaystyle \displaystyle\frac{a}{-c}=\frac{-a}{c}=-\frac{a}{c}$. In this case, $\displaystyle -(x-1)=1-x$.
Also, these two fractions are equal, not equivalent.
Or (more the same) just multiply both the numerator and the denominator by negative one:
$\displaystyle \displaystyle -\frac{1}{x-1} = \frac{-1}{x-1} = \frac{\left(-1\right)\left(-1\right)}{\left(-1\right)\left(x-1\right)} = \frac{1}{-x+1} = \frac{1}{1-x}$