Use of equivalent and equal

Question

Really simple question. Is valid to do this:

$$ 2x + 2 = 1 \iff x = \frac{1 - 2}{2} = -\frac{1}{2} $$

I mean is anything wrong when solved for $x$ to not use $\iff$ symbol more and just use $=$.

Answer 1

What you write is correct. I think it would be even better (equally correct and easier to read) with words:

$$ 2x + 2 = 1 $$ implies $$ x = \frac{1-2}{2} = -\frac{1}{2} . $$

Note the period at the end of the sentence - since it is a sentence.

In this case you don't care about the reverse implication, which happens to be true here but isn't always when manipulating expressions.

Replacing the $\iff$ by $=$ would make your argument just plain wrong - in a way all too commonly seen. Read literally it would say $$ 2x + 2 = 1 = \cdots = -\frac{1}{2} . $$

Answer 2

The equivalence sign is actually pretty smart since many students often write $$ 2x+2=1 \Longrightarrow x=-\dfrac{1}{2} $$ and ignore the fact that this is only a necessary condition. Observing that both directions, $\Longrightarrow $ AND $\Longleftarrow $ hold makes it unnecessary to check whether your finding $x=-1/2$ is actually a solution. This may be obvious in a case like this, but it isn't if you divided by a variable and haven't ruled out that it could have been zero, or if you square an equation. E.g. $x+1=0 \Longrightarrow x^2=1 \not\Longrightarrow x=1.$ The more complicated your algebraic manipulations are, the more is it necessary to check whether something found at the end of the calculations is actually a solution to the problem. By writing $\Longleftrightarrow $ you make sure that you checked the opposite direction, too.