Simple inequation with $x^2$ I was wondering what is the mathematics operations for :
$$x^2 > 9 \iff x > 3 \vee x < -3$$
I know that we can do that :
$$x^2 - 9 > 0$$
$$(x-3) (x+3) > 0$$
So : $$x - 3 > 0 \text{ and }  x + 3 > 0$$
Or, $$x - 3 < 0 \text{ and }  x + 3 < 0$$
So the solutions are $x > 3$ or $x > -3$.
But I was wondering if there is any mathematic operation for passing for $x^2 > 9$ to the solutions, without using my solution and without telling me that this is logic, or that we have done my solution on our mide and we just put the solutions "like a robot" ?
Can you help me ? :)
 A: You only need to be convinced that if $(x,y)$ are two real numbers then :
$$ xy > 0 \; \Leftrightarrow \; \big( x >0 \; \mathrm{and} \; y > 0 \big) \; \mathrm{or} \; \big( x < 0 \; \mathrm{and} \; y < 0 \big) $$
(which means that $xy>0$ if and only if $x$ and $y$ have the same sign.)
Note that :
$$ x^{2} > 9 \Leftrightarrow (x-3)(x+3) > 0 $$
and use the previous result !
A: $x^2\gt 3^2\iff |x-0|\gt 3$, so, on the number line, the distance between $0$ and $x$ is greater than $3$.  
In general, for $a\gt 0$, $$\begin{align}x^2\gt a^2&\iff |x-0|\gt a\\&\iff \text{the distance between $0$ and $x$ is greater than $a$}\\&\iff x\lt-a\ \ \text{or}\ \ x\gt a.\end{align}$$ 
$$\begin{align}x^2\lt a^2&\iff |x-0|\lt a\\&\iff \text{the distance between $0$ and $x$ is smaller than $a$}\\&\iff -a\lt x\lt a.\end{align}$$
Here, note that $|x-0|$ represents the distance between $0$ and $x$.
A: Here's another way of doing it
$$ x^2\gt 9 $$
$$ \sqrt{x^2}\gt \sqrt{9} $$
$$ |x|\gt 3$$
Therefore, every $x\in (-\infty, -3)\cup (3, \infty)$ is a solution.
To answer your question directly, one possible "robot" is Wolfram Alpha. However I'd suggest a pad and a pen. 
A: Since the function $f(x)=x^2$ is an even function, which means that $f(x)=f(-x)$. Because of this symmetry we can conclude, that if inequality 
$$x^2>9$$
answer is $x\in (3,\infty)$, then $x\in (-\infty ,-3)$ is also the answer. Same type of reasoning can be applied to other even functions, like $f(x)=x^4$ or $f(x)=\cos (x)$.
