TITLE: Symbol for unknown relation?
QUESTION [21 upvotes]: When solving equations like
$$\begin{align} 4x-4 &=\frac{(2x)^2}{x} \\
-4 &= \frac{4x^2}{x} -4x \\
-4 &= 4x -4x \\[0.2em]
-4 &= 0\end{align}$$
using the equality-symbol feels like abuse of notation, since you'll end up with $-4=0$, which is not an equality. For instance I feel it would be better to write
$$\begin{align} 4x-4 &\:\Box\:\frac{(2x)^2}{x} \\
-4 &\:\Box\: \frac{4x^2}{x} -4x \\
-4 &\:\Box\: 4x -4x \\[0.4em]
-4 &\:\Box\: 0 \\[0.3em]
-4 &\neq 0\end{align}$$
So I was wondering if there's a symbol or any other notations being used when trying to solve such an equation where you don't know if there's an equality?

REPLY [8 votes]: I like to put the $\iff$ (if and only if) symbol at the beginning of every new line, like this: $$\begin{align} 4x-4 &=\frac{(2x)^2}{x} \\
\iff-4 &= \frac{4x^2}{x} -4x \\
\iff-4 &= 4x -4x \\[0.2em]
\iff-4 &= 0\end{align}$$
So $4x-4=\frac{(2x)^{2}}{x}$ if and only if $-4=0$, which is true.