I can't solve $x-2<\frac{1}{x-1}$ We have to solve this inequation. 
$x-2<\frac{1}{x-1}$.
So we try $(x-1)(x-2)<1$ what's wrong ? 
Please, give me a hint.

$$x-2<\frac { 1 }{ x-1 } \\ x-2-\frac { 1 }{ x-1 } <0\\ \frac { { x }^{ 2 }-3x+1 }{ x-1 } <0\\ \frac { \left( x-\frac { 3-\sqrt { 5 }  }{ 2 }  \right) \left( x-\frac { 3+\sqrt { 5 }  }{ 2 }  \right) \left( x-1 \right)  }{ { \left( x-1 \right)  }^{ 2 } } <0\\ \left( x-\frac { 3-\sqrt { 5 }  }{ 2 }  \right) \left( x-\frac { 3+\sqrt { 5 }  }{ 2 }  \right) \left( x-1 \right) <0\\ x\in \left( -\infty ,\frac { 3-\sqrt { 5 }  }{ 2 }  \right) \cup \left( 1,\frac { 3+\sqrt { 5 }  }{ 2 }  \right) \\ $$
