$\displaystyle \frac{x}{x+8}=\frac{2}{3}$
The answer is $\displaystyle x=16$ but I can't see how they're coming to this answer...
The reason for "$\displaystyle x\ne -8$" is that, in that case, the fraction on the left is not defined.
And you go from one equation to the other by multiplying both sides by 3(x+ 8), the product of the two denominators.