1. (17 points) Solve the equation ||$|x-2|+3|-4|=-3 x-9$.


Answer: -5.

Solution. The set of possible solutions for the inequality $-3 x-9 \geq 0$, that is, $x \leq-3$. Under this condition, the inner absolute value is uniquely determined. Using the property of absolute value $|-a|=|a|$, we get the equation $||x-5|-4|=-3 x-9$. Similarly, we expand the inner absolute value, obtaining the equation $|x-1|=-3 x-9$. Under the obtained restriction, the last absolute value is uniquely determined, resulting in the equation $-x+1=-3 x-9$, that is, $x=-5$.