Is this interval notation for the solution of an inequality problem correct? Is this notationally correct?
$$
\begin{align*}
                    5 - x^2 &< -2, \\[1ex]
                    5 - x^2 + (-5) &< -2 + (-5), \\[1ex]
                    -x^2 &< -7, \\[1ex] 
                    (-x^2)(-1) &> (-7)(-1), \\[1ex] 
                    x^2 &> 7, \\[1ex]
                    x^2 - 7 &> 0,  \\[1ex]
                    (x + \sqrt{7})(x - \sqrt{7}) &> 0, \\[1ex]
                    x < -\sqrt{7} \,\lor\, x &> \sqrt{7}.
\; \llap{\mathrel{\boxed{\phantom{\;x < -\sqrt{7} \,\lor\, x > \sqrt{7}.}}}}\end{align*}
$$
or would the following be better notation?
$$
\begin{align*}
                    (x < -\sqrt{7}) \,\lor\, (x &> \sqrt{7}).
\; \llap{\mathrel{\boxed{\phantom{\;(x < -\sqrt{7}) \,\lor\, (x > \sqrt{7}).}}}}
\end{align*}
$$
Thank you.
 A: Your solution is correct, and so are both notations you used for the final result. There is no ambiguity in the first notation, because $\lor$ connects logical statements: it doesn't connect $-\sqrt{7}$ and $x$ because they are not statements, since you can't say they are true or false. So the second notation is just better to read but not really necessary.
However, as others pointed out, this is non-standard notation. Prefer to write the solution in natural language, as in "$x<-\sqrt{7}$ or $x>\sqrt{7}$", or in interval notation, that is, $x \in(-\infty, - \sqrt{7}) \cup (\sqrt{7}, \infty) $.
A: Your solution, which looks good to me, would also commonly be written in interval notation using the union operation:
$$(-\infty,-\sqrt{7})\cup (\sqrt{7},\infty)$$
A: I suggest either spelling out that logical symbol (for readability and because some readers may not know what it means) $$x < -\sqrt{7} \quad\text{or}\quad x > \sqrt{7},$$ or actually adopting interval notation $$(-\infty,-\sqrt{7})\cup (\sqrt{7},\infty)$$ as suggested by paw88789.
A: I will write the final part in question as $x\in (-\infty,-\sqrt7)\cup (\sqrt 7,\infty)$ because the inequality is in $x$.
