8. Find all values of the parameter $a$ for which the system

$$
\left\{\begin{array}{l}
y=\frac{|x|-1}{|x+1|}-2 \\
4|x-1.5-a|+|y-1-2a|=1
\end{array}\right.
$$

has a unique solution. In your answer, specify the smallest of all the obtained values of the parameter $a$.

points
Solution. We will construct the graphs of the functions:

a) $y=\frac{|x|-1}{|x+1|}-2=\left\{\begin{array}{l}-1, \text { for } x<-1, \\ -3, \text { for }-1<x \leq 0, \\ \frac{x-1}{x+1}-2=\frac{-x-3}{x+1}=-1-\frac{2}{x+1}, x \geq 0,\end{array}\right.$

b) $4|x-1.5-a|+|y-1-2a|=1$ - a rhombus, the center of which moves along the line $y=2x-2$.

![](https://cdn.mathpix.com/cropped/2024_05_06_58f34d7e66d046d9b7d5g-05.jpg?height=526&width=420&top_left_y=899&top_left_x=1503)

Common points of the graphs exist only for $-1<x \leq 0$ - in this case, we get only one intersection at the point $(0,-3), a=-1.5$.

Answer: -1.5.