# How are these two equivalent

• Nov 24th 2010, 04:37 AM
Hellbent
How are these two equivalent
Hi,

I'm not seeing how these two are equivalent: $-\frac{1}{x-1} \equiv \frac{1}{1-x}$.

• Nov 24th 2010, 04:45 AM
emakarov
We have $a\cdot(-b)=(-a)\cdot b=-ab$. In particular, when $b = 1/c$, we have $\displaystyle\frac{a}{-c}=\frac{-a}{c}=-\frac{a}{c}$. In this case, $-(x-1)=1-x$.

Also, these two fractions are equal, not equivalent.
• Nov 24th 2010, 04:52 AM
TheCoffeeMachine
Or (more the same) just multiply both the numerator and the denominator by negative one:

$\displaystyle -\frac{1}{x-1} = \frac{-1}{x-1} = \frac{\left(-1\right)\left(-1\right)}{\left(-1\right)\left(x-1\right)} = \frac{1}{-x+1} = \frac{1}{1-x}$
• Nov 24th 2010, 05:56 AM
Wilmer
x = 3

-(1 / (3 - 1)) = -(1 / 2) = -1/2

1 / (1 - 3) = 1 / -2 = -1/2