Task B-1.1. Solve the equation:

$$
|x-| x-|x+1|||=x
$$
## Solution.

Since $x$ is equal to the left side of the equation which is non-negative, it must be $x \geq 0$.

Therefore, $x+1 \geq 0$. It follows that

$$
x=|x-| x-(x+1)||=|x-|-1||=|x-1| \text {. }
$$

Now we have two cases:

(a) $x-1 \geq 0$, i.e., $x \geq 1$. Then $x=x-1$ from which it follows that $0=-1$, which means there is no solution in this case.

(b) $x-1<0$, i.e., $0 \leq x<1$ (due to the initial condition that $x$ is non-negative). Then $x=-x+1$ from which we get the solution $x=\frac{1}{2}$, which satisfies the condition.