## Task B-1.2.

Solve the equation

$$
|| x-2023|+2022 x|=|2022-| 2023-x|| .
$$
## Solution.

Two real numbers have the same absolute value if and only if they are either equal or opposite numbers. Therefore, it holds that

$$
|x-2023|+2022 x=2022-|2023-x| \text { or }|x-2023|+2022 x=-2022+|2023-x| \text {. }
$$

Note that $|x-2023|=|2023-x|$. In the first case, we get the equation

$$
\begin{aligned}
2|x-2023| & =2022-2022 x, \text { i.e., } \\
|x-2023| & =1011-1011 x
\end{aligned}
$$

This equation will have a solution if $1011-1011 x \geqslant 0$, i.e., if $x \leqslant 1$. Then we have either

$$
\begin{aligned}
x-2023 & =1011-1011 x \\
1012 x & =3034 \\
x & =\frac{3034}{1012}>1,
\end{aligned}
$$

which is not a solution, or

$$
\begin{aligned}
x-2023 & =-1011+1011 x \\
-1010 x & =1012 \\
x & =-\frac{1012}{1010}=-\frac{506}{505}
\end{aligned}
$$

which is a solution.

In the second case, from $|x-2023|+2022 x=-2022+|2023-x|$ we get $2022 x=-2022$, so $x=-1$.

All solutions to the given equation are $x=-\frac{506}{505} \text{ and } x=-1$.