$8.56 \frac{1}{\log _{2}(x-1)}<\frac{1}{\log _{2} \sqrt{x+1}}$.
8.56 Let the inequalities $00
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From (1) and (2) it follows that for $11$, i.e., $x>2$. Then both logarithms are positive and we arrive at the inequality $\log _{2} \sqrt{x+1}3$.

Answer: $(1,2) \cup(3, \infty)$.