Comparison issue I'm given: $\frac 12x -1 = \frac23x + \frac12$.
I'm aware that I should apply the rule where one either divides or multiplies, this time I'm told to multiply by 6. 
$$\left(\frac 12x -1\right) \times 6 = \left(\frac23x + \frac12\right)\times6$$
$$\frac 62x -6 = \frac{12}{3}x + \frac62$$
$$3x-6=4x+3$$
$-x=9$ so: $x=-9$.
My question:
Why do we choose 6 to multiply? Where do we get 6 from?

In my view it is better to start by putting the terms (summands) with the variable $x$ on one side and leave the rest on the other side.
$\frac 12x -1 = \frac23x + \frac12$.
First we add 1 on both sides to get rid of the -1 on the left side. 
$\frac 12x = \frac23x + \frac12+1$.
Substracting $\frac23x$ on both sides
$\frac 12x - \frac23x=  \frac12+1$.
Factoring out x on the LHS
$x\cdot \left(\frac 12 - \frac23 \right)=  \frac32$
Now we take the Least common multiple (LCM) of number in the denominators 2 and 3 to sum up the fractions in the brackets on the LHS. $LCM(2,3)=6$. Thus we get
$x\cdot \left(\frac 36 - \frac46 \right)=  \frac32$
$x\cdot \left( - \frac16 \right)= \frac32$
Multiplying both sides by $-6$
$x\cdot \underbrace{\left( - \frac16 \right)\cdot (-6)}_{=1}= \frac32\cdot (-6)$
$x=-\frac{18}{2}=-9$
