Solve $x>\sqrt{1-x}$

Question

The solution to the above question: https://i.stack.imgur.com/JhG4O.jpg

In illustration 1.5, I can say that $x$ must be positive after plugging in negative values for $x$ and observing that the inequality isn't satisfied, but I do not understand how to logically conclude that $x$ must be positive. Is there an alternate way to prove that $x$ is positive other than experimentation?

Answer 1

Hint: $\sqrt{t} \geqslant 0$ by definition.

Answer 2

Hint: The $\sqrt{1-x}$ function always gives non-negative values. And $x$ is strictly greater than $\sqrt{1-x}$.