Question

Solve:
$\frac{1}{x-2}+\frac{2}{x-1}=\frac{6}{x}$ where $x\ne 1,2$

Answer 1

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

$\Rightarrow$ $\frac{x-1+2(x-2)}{(x-2)(x-1)}=\frac{6}{x}$

$\Rightarrow$ $\frac{x-1+2x-4}{x^2-x-2x+2}=\frac{6}{x}$

$\Rightarrow$ $\frac{3x-5}{x^2-3x+2}=\frac{6}{x}$

$\Rightarrow$ $3x^2-5x=6x^2-18x+12$

$\Rightarrow$ $3x^2-13x+12=0$

$\Rightarrow$ $3x^2-9x-4x+12=0$

$\Rightarrow$ $3x(x-3)-4(x-30=0$

$\Rightarrow$ $(x-3)(3x-4)=0$

$\Rightarrow$ $x-3=0$ and $3x-4=0$

$\therefore$ $x=3$ and $x=\frac{4}{3}$