Factoring fractions to solve for X

[tomp]

In a GRE sample question:
x/3 - x/6 + x/9 - x/12 = 1 - 1/2 + 1/3 - 1/4, the answer "x = 3" was found by "factoring x/3 out of the left side of the expression" making it x/3 - x/6 + x/9 - x/12 = x/3 (1 - 1/2 + 1/3 - 1/4).
Supposedly, the expression should then read: x/3 (1 - 1/2 + 1/3 - 1/4) = 1 - 1/2 + 1/3 - 1/4, at which point both sides cancel out, leaving x/3 = 1, or x = 3.

Can someone explain how multiplying the right side by x/3 is "factoring"? I just don't get it! Where did all the other "x" fractions on the left side go?

Can anyone explain the procedure used? Thanks!

[Math Tutor]

[tex]\frac{x}{3} -\frac{x}{6} + \frac{x}{9} - \frac{x}{12} =[/tex]

[tex]= \frac{x}{3} - \frac{x}{3}.\frac{1}{2} + \frac{x}{3}.\frac{1}{3} - \frac{x}{3}.\frac{1}{4} =[/tex]


[tex]= \frac{x}{3}(1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4})[/tex]

Is it clear now?

[tomp]

Thanks very much for replying!
The multiplication is clear, but I still can't figure out why everything on the right side is arbitrarily being multiplied by x/3. Wouldn't we have to do that to everything on both sides of the = sign to keep the equation consistent?
Sorry for my thick-headedness, but if you could clue me in on the logic behind what's being done I'd be most grateful!
p.s. I've attached a jpeg of the original problem

[Math Tutor]

They show you the fastest and the most difficult way.

I would like to advise you not to follow that way.
Better to make all calculations on the both sides of the equation.

The most important is to find the right answer no matter the way.

[dduclam]

[tex]\frac{x}{3} -\frac{x}{6} + \frac{x}{9} - \frac{x}{12} =[/tex]

[tex]= \frac{x}{3} - \frac{x}{3}.\frac{1}{2} + \frac{x}{3}.\frac{1}{3} - \frac{x}{3}.\frac{1}{4} =[/tex]


[tex]= \frac{x}{3}(1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4})[/tex]

Is it clear now?

[tex]LSH= x(\frac{1}{3} -\frac{1}{6} + \frac{1}{9} - \frac{1}{12})=x.\frac7{36}[/tex]

[tex]RSH= 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4}=\frac7{12}[/tex]

[tex]LSH=RSH <=> x.\frac7{36}=\frac7{12} <=>x=3[/tex]

[Guest]

Here is a step by step solution:

http://www.symbolab.com/solutions/quadr ... 0-%201%2F4