Example 6. Solve the equation

$$
\frac{1-2 x}{3-|x-1|}=1
$$
Solution. The given equation is equivalent to the combination of two systems:

![](https://cdn.mathpix.com/cropped/2024_05_21_aca6fef227914c05e0acg-039.jpg?height=182&width=813&top_left_y=777&top_left_x=630)

Solving the equation $\frac{1-2 x}{4-x}=1$, we find $x_{1}=-3$ - its only root. However, it does not satisfy the condition $x-1 \geqslant 0$; therefore, the first system of the combination has no solutions.

Solving the equation $\frac{1-2 x}{2+x}=1$, we find $x_{2}=-1 / 3$ - its only root. It satisfies the condition $x-1<0$, and therefore the number $-1 / 3$ is a solution to the second system of the combination.

Thus, the only solution to the original equation is the number $-1 / 3$.

When solving an equation where the expression under the modulus sign also contains a modulus, one should first eliminate the inner moduli, and then open the remaining moduli in the obtained equations.