#### Answer

$x=-1$

#### Work Step by Step

$y=\dfrac{x+6}{3x-12}-\dfrac{5}{x-4}-\dfrac{2}{3}$
Set $y$ equal to $0$:
$\dfrac{x+6}{3x-12}-\dfrac{5}{x-4}-\dfrac{2}{3}=0$
Factor the denominator of the first fraction:
$\dfrac{x+6}{3(x-4)}-\dfrac{5}{x-4}-\dfrac{2}{3}=0$
Multiply the whole equation by $3(x-4)$:
$3(x-4)\Big[\dfrac{x+6}{3(x-4)}-\dfrac{5}{x-4}-\dfrac{2}{3}=0\Big]$
$(x+6)-(5)(3)-2(x-4)=0$
$x+6-15-2x+8=0$
Take all terms without $x$ to the right side, simplify and solve for $x$:
$x-2x=-8-6+15$
$-x=1$
$x=-1$