1. (17 points) Solve the equation ||$|x-1|+2|-3|=-2 x-4$.


Answer: -4.

Solution. The set of possible solutions for the inequality $-2 x-4 \geq 0$, that is, $x \leq-2$. Under this condition, the inner absolute value is uniquely determined. Using the property of the absolute value $|-a|=|a|$, we get the equation $||x-3|-3|=-2 x-4$. Similarly, we expand the inner absolute value, obtaining the equation $|x|=-2 x-4$. Under the obtained restriction, the last absolute value is uniquely determined, resulting in the equation $-x=-2 x-4$, that is, $x=-4$.