4. Find all values of the parameter $a$, for each of which the equation $a|2-x|+\frac{x^{2}-x-6}{3-x}=0$ has exactly one solution
Answer: $a \in(-1 ; 1] \cup\{5\}$.

Solution. Given the condition $x \neq 3$, the equation is equivalent to $a|x-2|=x+2$. The graph of the right side of the equation is the line $y=x+2$. The graph of the left side of the equation is a "V" shape with its vertex at the point $(2 ; 0)$, and the slope of its branches is determined by the parameter $a$.

The left branch of the "V" intersects the line when $a>-1$, and the right branch intersects the line when $a>1$ and $a \neq 5$ (when $a=5$, the right branch passes through the punctured point of the line - the point $(3 ; 5)$). Therefore, exactly one solution is obtained when $a \in(-1 ; 1] \cup\{5\}$.