1. Find all real numbers $x$ that satisfy the equation

$$
|2006-| 206-x||=26
$$
I/1. Since $|2006-| 206-x||=26$, it must be that $2006-|206-x|= \pm 26$. If $2006-|206-x|= 26$, then $|206-x|=1980$, which gives $x=206 \pm 1980$. If $2006-|206-x|=-26$, then $|206-x|=2032$, which gives $x=206 \pm 2032$. Therefore, all solutions are $-1774, 2186, -1826$, and 2238.

![](https://cdn.mathpix.com/cropped/2024_06_07_fb5898f0472fc84ca82bg-06.jpg?height=211&width=1639&top_left_y=1005&top_left_x=217)