4. (20 points) Determine the number of solutions of the equation $a(x-|x|+2)=|x-3|-2$ for each value of the parameter $a$.
Solution. We will open the modules and investigate three cases separately:

1) $x \in(-\infty ; 0) a(2 x+2)=1-x ; x(2 a+1)=1-2 a$. When $a=-0.5$, the equation has no solutions for these x. Divide by the coefficient, $x=\frac{1-2 a}{1+2 a}$. Solve the inequality $\frac{1-2 a}{1+2 a} < 0 ; a > -1 ; a \in(-1 ;+\infty)$ - one solution; $a \in(-\infty ;-1]$ - no solutions. Summarizing the obtained results, we get the answer.

This problem can also be solved graphically in the Oxa coordinate system.

Answer: For $a \in(-\infty ;-1)$ - one solution; for $a=-1$ - two solutions; for $a \in(-1 ;-0.5)$ - three solutions; for $a \in[-0.5 ;+\infty)$ - two solutions.

| Content of the criterion | Points |
| :--- | :---: |
| The correct answer is obtained and justified. | 20 |
| The answer differs from the correct one by one or two points (extreme in intervals or separate). | 15 |
| The modules are correctly opened and the number of solutions is investigated in each of the three cases, but the results are incorrectly correlated; or an error in one of the cases of opening the modules leads to an incorrect answer. | 10 |
| The modules are correctly opened and at least one of the cases is correctly investigated, but there is no further solution or it is incorrect. | 5 |
| The solution does not meet any of the criteria listed above. | 0 |
| Maximum score | 20 |