Is there a reason to omit $[0]$ from the solution of the inequality $x^4 \ge x^2$?

Question

This is the inequality I plugged in: $$x^4 \ge x^2$$ Wolfram Alpha gives me the correct number line, but the interval notation it gives me is missing $\{0\}$. (It gives me $(-∞, -1]$ and $[1, ∞)$, missing $\{0\}$).

You can test this out if you want. Is there something I'm not seeing, or is Wolfram Alpha wrong?

Answer 1

You're right, there is something wrong or at least misleading with the specific subsection "Intervals" of WolframAlpha's output (it does not give all the solutions). Note that the other sections (number line) are right, however. You can also try with Mathematica's syntax, which also gives the right answer:

Reduce[{x^4 >= x^2}, x, Reals]

You can also use WolframAlpha with more specific inputs: for instance, if you specify "solve"

solve x^4 >= x^2

then the result is much less ambiguous.

Answer 2

It gives me $[1, ∞) \cup (-∞, -1]$

No it doesn't. It gives $[1, \infty)$ and it gives $(-\infty, -1]$. That is, it gives some intervals on which the inequality holds. Also note, that there is a point "solutions" and then also "solution."

I tend to agree that this behavior is not very intuitive, but I do not think there is an actual error there, and in any case you misrepresent the output.