Example 17. Solve the inequality

$$
(x-2)^{x^{2}-6 x+8}>1
$$

## 170 CH. 3. INEQUALITIES WITH ONE UNKNOWN
Solution. The domain of admissible values for this inequality is defined by the condition $x>2$. For such $x$ we have

$(13) \Leftrightarrow 10^{\left(x^{2}-6 x+8\right) \lg (x-2)}>1 \Leftrightarrow\left(x^{2}-6 x+8\right) \lg (x-2)>0 \Leftrightarrow$

$$
\begin{aligned}
& \Leftrightarrow\left[\begin{array}{l}
\left\{\begin{array} { l } 
{ x ^ { 2 } - 6 x + 8 > 0 , } \\
{ \operatorname { l g } ( x - 2 ) > 0 , } \\
{ \{ \begin{array} { l } 
{ x ^ { 2 } - 6 x + 8 < 0, \\
\lg (x-2)<0, \\
(x-2)(x-4) < 0, \\
2 < x < 4, \\
\end{array} } \\
{ \{ \begin{array} { l } 
{ 2 < x < 3, } \\
{ x - 2 > 1, } \\
{ x > 3, } \\
\end{array} } \\
{ \{ \begin{array} { l } 
{ 2 < x < 4, } \\
{ x > 3, } \\
{ 2 < x < 3, } \\
\end{array}\right.\right.\right.
\end{aligned}
$$

Thus, the set of all solutions to the inequality (13) consists of the union of two intervals: $(2 ; 3) \bigcup(4 ;+\infty)$.