Question

Solve the following inequality:
$1+\frac{2}{x-1}>\frac{6}{x}$

Answer 1

$1+\frac{2}{x-1}>\frac{6}{x}$

$1+\frac{2}{x-1}-\frac{6}{x}>0$

$\frac{x(x-1)+2x-6(x-1)}{x(x-1)}>0$

$\frac{x^2-x+2x-6x+6}{x(x-1)}>0$

$\frac{x^2-5x+6}{x(x-1)}>0$

$\frac{x^2-3x-2x+6}{x(x-1)}>0$

$\frac{(x-3)(x-2)}{(x-0)(x-1)}>0(x=4,\frac{1\times 2}{4\times 3}>0)$

$x\epsilon (-\infty ,0)\bigcup (1,2)\bigcup (3,\infty )$