8. $\left\{\begin{array}{l}\log _{3}(a+2 x) \leq 1 \\ |a-x| \leq 2\end{array} \Leftrightarrow\left\{\begin{array}{l}0<a+2 x \leq 3 \\ -2 \leq x-a \leq 2\end{array} \Leftrightarrow\left\{\begin{array}{l}-\frac{a}{2}<x \leq \frac{3-a}{2} \\ a-2 \leq x \leq a+2\end{array}\right.\right.\right.$

To illustrate the solutions of the obtained system on a number line with shading:

![](https://cdn.mathpix.com/cropped/2024_05_06_62b84f4e75df73adcd4cg-8.jpg?height=281&width=860&top_left_y=2069&top_left_x=447)

For the system to have a unique solution, it is necessary and sufficient that the left boundary of the interval $[a-2 ; a+2]$ coincides with the right boundary of the half-interval $(-a / 2 ; (3-a) / 2]$, i.e., $(3-a) / 2 = a-2$; $a = 7 / 3$ and $x = 1 / 3$.
Answer: $a=7 / 3$ and $x=1 / 3$.

Answers.

|  | 1st option | 2nd option | 3rd option | 4th option |
| :---: | :---: | :---: | :---: | :---: |
| 1 | possible | possible | possible | possible |
| 2 | 27 | 8 | 40 | 1 |
| 3 | 6 | $56 / 3$ | $32 / 3$ | 10.5 |
| 4 | $(15 ; 18)$ | $(5 ; 3 ; 19)$ | $(3 ; 3 ; 2)$ | $(0 ; 0)$  $(-2 ; 0)$ |
| 5 | 83 | 8 hours 45 min | 4 | 2 |
| 6 | 12 | 60 | 48 | $9 \sqrt{3}$ |
| 7 | $(\pi k ; \pi m ; \pi n)$ | $\frac{\pi k}{2} ; \frac{2 \pi n}{3}$  $\pm \frac{2 \pi}{3}+2 \pi m$ | $\frac{\pi m}{3} ; \frac{\pi n}{5}$ | $\frac{\pi}{2}+\pi n$  $\frac{\pi}{2}+\pi m$  $\frac{\pi}{2}+\pi k$ |
| 8 | $a=2 \sqrt{3}$  $(0 ;-1)$ | $a=6$  $x=-2, y=0$. | $a=3$  $x=1$ | $a=7 / 3$  $x=1 / 3$ |