Answer
The statement is false.
To make it true,
replace "can always" with "can not always".
Work Step by Step
No. Here is a counterexample
$\displaystyle \frac{x}{x-2}-\frac{1}{x-2}=\frac{x-1}{x-2}$
which has no common factors. Reducing (simplifying) can't be done here.
The statement is false.
To make it true,
replace "can always" with "can not always".
