When I write that $\frac{1}{x} - \frac{1}{x} = 0$, should I include the fact $x\neq0$? When I write that $\frac{1}{x} - \frac{1}{x} = 0$, should I include the fact $x\neq0$?
 A: Yes you have to write $x\neq 0$. But why? Well division is really just multiplication. By definition, $x/y:=x\cdot \frac{1}{y}$ and multiplication is an operation defined on $\mathbb{R}$. To be precise, is is the map $\cdot: \mathbb{R}\times \mathbb{R}\to \mathbb{R}$, $(x,y)\mapsto x\cdot y$. Therefore, in order to even talk about multiplying two elements, you need both to be in $\mathbb{R}$. But $\frac{1}{x}\not\in \mathbb{R}$ if $x=0$ (in fact it is not even defined). 
A: An equation is what is called a statement form; it includes a placeholder, here it's $x$.  
Now we call any number a solution if the statement which is produced by substituting the placeholder by that number is a statement which is true.
Take $2x=6$, e.g.  By substituting $x$ by $5$ the statement $2\cdot5=6$ is achieved.  But as that statement isn't true, $5$ is no solution of that statement form.
Now there may be substitutions which don't produce a statement at all, but rather something incomputable (something that is not a number).  Incomputability arises in our case if you plug in $x=0$ as $1/0$ is not a number.  
Hence $0$ is neither a solution nor not a solution of the given statement form.  In case you substitute any other number a statement which is true is produced.
