9.304 Find $x$, for which the inequality $y^{2}-\left(5^{x}-1\right)(y-1)>0$ holds for all $y$.
## Solution.

$y^{2}-\left(5^{x}-1\right) y+5^{x}-1>0$. This inequality holds for all $y$ if and only if $D<0$, where

$D=\left(5^{x}-1\right)^{2}-4 \cdot 5^{x}+4=\left(5^{x}-1\right)\left(5^{x}-5\right)<0 \Leftrightarrow 1<5^{x}<5 \Leftrightarrow 0<x<1$.

Answer: $x \in(0 ; 1)$.