Error: $x = 1\iff x = \pm 1$

What's the error in this equation/solution?

The correct answer should be $$x = 1 ⇒ x = ±1$$, right?

Here's my attempt:
For $$x=-1$$, I can't "come back" from the first line to $$x=1$$. I get $$4=0$$ as a result.
However, for $$x=1$$, I can do this procedure and find $$0=0$$.

All help is appreciated.

• $x^2-2x+1=0$ does not imply $x=1$? What other solution exists? – David C. Ullrich Mar 30 at 15:50
• My mistake, Sorry – Ray LittleRock Mar 30 at 15:54

$$(1)$$ and $$(2)$$ are not equivalent.
As you can see, $$(2)$$ also holds for $$x=-1$$.
• @DanielSehnColao If I understand you correctly then it looks like you get the idea. It's true that $x^2 = 1$ whenever $x=1$ or $x=-1$. Indeed the important part is that once they insert $x=1$ in $-2x$ they change everything because it has different values for $x=1$ and $x=-1$. – Yanko Mar 30 at 19:38
The answer given is right. The second logical connective (i.e., $$\iff$$) should be $$\implies$$.