Squared length triangle inequalities

Geometry Level 2

A triangle has side lengths $x,$ $x^2,$ and $\frac{1}{2}.$ Which of the following intervals describes all the values that $x$ can take?

\left(\frac{-1+\sqrt{3}}{2},\frac{1+\sqrt{3}}{2}\right)

\left(0,\frac{1+\sqrt{3}}{2}\right)

\left[\frac{1}{2},1\right]

\left(\frac{-1+\sqrt{3}}{2},\frac{1}{2}\right] \cup \left[1,\frac{1+\sqrt{3}}{2}\right)

\left(\frac{-1+\sqrt{3}}{2},\infty\right)