Symbol for unknown relation?

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?

Solution 1:

It's perfectly fine to have equality signs.

When you solve equations, what you really do is say

Assume that the following is true

$$4x-4=\frac{(2x)^2}{x}$$

then

$$-4=0$$

is true.

Contradiction, the original assumption is false.

Solution 2:

I put a question mark above an equal sign, like this: $\stackrel{?}=$. (In MathJax, just type $\stackrel{?}=$.)

Solution 3:

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.