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

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

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

points
Solution. We will solve the given system of equations graphically.

Let's plot the graphs of the functions a) $y=\frac{x+1}{|x|-1}$ and b) $|x+y+a|+|x-y-a|=1$.

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

For $x>1$ there are no intersections, since in this problem $|x| \leq 1 / 2$ always, which follows from the second equation. There are no intersections with the left branch for $y>-1$. Substituting $y=-1 \, (x<0)$, we get

![](https://cdn.mathpix.com/cropped/2024_05_06_58f34d7e66d046d9b7d5g-21.jpg?height=391&width=1719&top_left_y=1666&top_left_x=183)
$3 / 2$ is obtained as a solution in the form of an entire segment.

For $-3<y<-1$ there are two intersections, and for $y=-3$ there is a single common point $(0.5,-3)$.

In this case, the value of the parameter $a$ can be obtained by substituting the coordinates of the common point into the second equation of the system: $|-3+a|+|3-a|=1 \Rightarrow a=3+1 / 2=3.5$ (The second value 2.5 does not fit, since in this case there will be 2 solutions.)

Answer: 3.5.