Question

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

• A. $x=3,\frac{4}{3}$
• B. $x=4,\frac{3}{4}$
• C. $x=-4,\frac{3}{4}$
• D. $x=-3,\frac{3}{4}$

Answer 1

The correct option is A $x=3,\frac{4}{3}$

We have:

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

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

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

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

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

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

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

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

$\Rightarrow x-3=0or3x-4=0$

$\Rightarrow x=3orx=\frac{4}{3}$