I don't think I've ever mastered this concept. If I have a problem like $\displaystyle \frac{x+2a+3b}{x} = {4} $, can I just divide the x out to get $\displaystyle 1+2a+3b = 4 $ ?

I know if I look at the first expression like $\displaystyle \frac{x}{x} + \frac{2a}{x} + \frac{3b}{x} $ then I could turn x/x into 1, but the expression would look like $\displaystyle 1 + \frac{2a}{x} + \frac{3b}{x} $. The x could get brought back into the quotient if it had a common denominator.

Also, if the sample problem were $\displaystyle \frac{x*2b+a}{x} $ would that make a difference in mathematical rules for dividing out the x? (As opposed to addition or subtraction).

I hope my question makes sense, if not I can explain more.