Is it "$\Rightarrow$" or "$\Leftarrow$" in "$x^2 = 9 \text{ ? } x = 3$"?

Question

The equation $x^2 = 9$ has the solution set {-3,3}, that is $(x = -3 \vee x = 3)$. So I would understand that $$ x^2 = 9 \iff (x = -3 \vee x = 3) $$ Now "usually" (perhaps) one writes in course of solving an equation $$ \dots\quad x^2 = 9 \Rightarrow x = 3 $$ but I think that is wrong, because a set implies a "larger" or equal set, So the correct one should read $$ x^2 = 9 \Leftarrow x = 3 $$ So which one is correct ?

Answer 1

$$x^2=9\Rightarrow x=3$$ is wrong.

Try $x=-3.$ $$x=3\Rightarrow x^2=9$$ is true of course.

Also $x^2=9$ is equivalent to $x=3$ or $x=-3$ because $x^2-9=(x-3)(x+3).$

Answer 2

To begin with, $x^2=9 \Leftarrow x=3$ is certainly correct.

Regarding the other way around, $x^2=9 \Rightarrow x=3$ it depends. If $x$ earlier has been restricted to be non-negative, e.g. if $x \in \mathbb{N}$, then the implication is correct. If $x$ can be negative, the implication is not correct, since also $x=-3$ is a possibility.

Answer 3

$x^2 = 9 \iff x^2 - 9 = 0$ is clear. Of course $x^2 -9 = (x-3)(x+3)$, and in all fields (like $\Bbb R$ or $\Bbb C$) we have that $$ab= 0 \iff (a = 0) \lor (b=0)\tag{1}$$ for all $a,b$ in that field.

So $$x^2 = 9 \iff (x-3)(x+3)= 0 \iff (x-3 = 0) \lor (x+3=0) \iff x=3 \lor x= -3$$

So every step is reversible by standard facts about inverses and $(1)$.