5. Solve the inequality $\left(2^{x^{2}-6}-4 \cdot 2^{x+4}\right) \log _{\cos \pi x}\left(x^{2}-2 x+1\right) \geq 0$. (10 points)
# Solution:

Domain: $(x-1)^{2}>0, \quad \cos \pi x>0, \cos \pi x \neq 1, \Rightarrow x \in(-0.5+2k ; 2k) \cup(2k ; 0.5+2k), k \in Z$. The original inequality on the domain is equivalent to the following $\left(x^{2}-x-12\right)(\cos \pi x-1)\left(x^{2}-2 x\right) \geq 0 \Leftrightarrow(x-4)(x+3)(x-2) x \leq 0 \Rightarrow x \in[-3 ; 0] \cup[2 ; 4]$.

Considering the domain, we have $\quad x \in(-2.5 ;-2) \cup(-2 ;-1.5) \cup(-0.5 ; 0) \cup(2 ; 2.5) \cup(3.5 ; 4)$.

Answer: $x \in(-2.5 ;-2) \cup(-2 ;-1.5) \cup(-0.5 ; 0) \cup(2 ; 2.5) \cup(3.5 ; 4)$.