solution of the inequality$ x>(1-x)^{1/2}$ is given by

Question

How do I approach the soltuion(s) of the inequality $$f(x)= x>(1-x)^{1/2}$$

I have tried my best to do it but still unable to solve it.

Answer 1

$x>0$ and the domain gives $1-x\geq0.$

Thus, $0<x\leq1$ and we have $$x^2>1-x,$$ which gives $x>\frac{-1+\sqrt{5}}{2}$ and we got the answer:

$$\left(\frac{\sqrt5-1}{2},1\right]$$

Answer 2

Hint:

As for real $x,$ $$\sqrt{1-x}\ge0, x>\sqrt{1-x}\ge0$$

In that case,

$$x>\sqrt{1-x}\implies x^2>1-x$$