Answer
$$x ∈ (-\infty, -0.53] ∪ [0.53, +\infty)$$
Also see the image below.
Work Step by Step
$\sqrt{0.5x^2+1}\leq 2|x|$
$\sqrt{0.5x^2+1}$ is represented as the Blue graph on the image above.
$2|x|$ is the Red graph on the image above.
We need to find the region, where the Blue graph is below the Red graph. As we see from the image, these regions are:
$(-\infty, -0.53] ∪ [0.53, +\infty)$