Nonchalant Cancellation

$\dfrac{A}{A+B}>\dfrac{B}{A+B}\\ \dfrac{A-B}{A+B}>0$

We know that $A>B \implies A-B>0$

The numerator of the fraction will always be positive.

Therefore, the inequality holds when the denominator is positive as well:

$A+B>0 \implies A>-B$

$A>B$ does not guarantee that $A>-B$ . For example,

$A=-1,\;B=-2\implies A>B$ , but at the same time, $A<-B$ .

For this inequality to always hold true, $A>0$ , which is not stated.

Therefore, the answer is $\boxed{\text{False}}$

For $A= 1$ and $B =2,$

$\large \frac 13 \not > \frac23.$

So the given inequality is not always true.