8. Find all values of the parameter $a$ for which the system $\left\{\begin{array}{l}y=\frac{x+1}{|x|-1}, \\ |x+y|+|x-y-2 a|=1 .\end{array}\right.$

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) $|x+y|+|x-y-2a|=1$.

![](https://cdn.mathpix.com/cropped/2024_05_06_58f34d7e66d046d9b7d5g-14.jpg?height=405&width=479&top_left_y=320&top_left_x=183)

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

For $x>1$, there are no common points. With the left branch, there is only one common point $(0,-1)$. The value of the parameter $a$ can be obtained by substituting the coordinates of the common point into the second equation of the system: $|-1|+|1-2a|=1 \Rightarrow a=1/2$.

2nd method. Consider $x<0$, in which case $y=-1$. Notice that in this problem, $|x+y| \leq 1$ always, which follows from the second equation. Substituting $y=-1$, we get $|x-1| \leq 1 \Rightarrow -1 \leq x-1 \leq 1 \Rightarrow 0 \leq x \leq 2$, which is impossible due to the negativity of $x$. Consider $x \geq 0$. In this case, $y=\frac{x+1}{x-1}=1+\frac{2}{x-1}$. Substituting into the first modulus of the second equation, we get $-1 \leq x+1+\frac{2}{x-1} \leq 1 \Rightarrow -2 \leq \frac{x^2-x+2}{x-1} \leq 0$. The numerator of the fraction is always positive, so the fraction can be less than zero only when $x<1$.

Then $\left\{\begin{array}{l}0 \leq x<1, \\ -2 \leq \frac{x^2-x+2}{x-1},\end{array} \Rightarrow\left\{\begin{array}{l}0 \leq x<1, \\ 0 \leq \frac{x^2-x+2+2x-2}{x-1},\end{array} \Rightarrow\left\{\begin{array}{l}0 \leq x<1, \\ 0 \leq \frac{x^2+x}{x-1},\end{array} \Rightarrow x=0\right.\right.\right.$.

In this case, $y=-1$, substituting the coordinates of the point into the second equation of the system, we find the parameter: $|-1|+|1-2a|=1 \Rightarrow a=1/2$.

Answer: 0.5.