Determine the intervals in which the following inequality is satisfied: $|1-x|-x\geq 0$ Exercise:

Determine the intervals in which the following inequality is satisfied: $$|1-x|-x\geq 0$$


Attempt:
What to Expect:
A quick manipulation renders the following: $|1-x|\geq x$.
Graphing both sides:

Eyeballing, the answer seems to be: $x \leq \frac{1}{2}$.
Solution:
(1) $|1-x|-x\geq 0$
(2) $|1-x| \geq x$
(3)


*

*when $x \geq 0$: 


*

*when $1-x \geq 0$: $1-x \geq x$

*when $1-x < 0$: $-(1-x) \geq x$


*when $x < 0$: $x-1 > 0$


(4)


*

*when $x \geq 0$: 


*

*when $1 \geq x$: $\frac{1}{2} \geq x$

*when $1 < x$: invalid


*when $x < 0$: $x > 1$



Request:
I do see the expected answer in (4), but according to my solution it's only applicable when $x \geq 0$. When $x < 0$, I get an answer that seems to have no resemblance in the expected answer. Where and what did I do wrong?

When $x \ge 1$,
$$|1-x|=x-1\ge x$$
and there is no solution.
When $x \le 1$,
$$|1-x|=1-x\ge x$$
$$1\ge 2x$$
$$1/2\ge x$$
