Answer
$x=-2$
Work Step by Step
Rewrite the equation:
$$\frac{1}{(x+6)(x-3)}-\frac{3}{x+6}-\frac{4}{x-3}=0$$
$$\frac{1-3(x-3)-4(x+6)}{(x+6)(x-3)}=0$$
$$1-3x+9-4x-24=0$$
$$-14-7x=0$$
$$-14=7x$$
$$7x=-14$$
$$x=\frac{-14}{7}$$
$$x=-2$$
Checking:
$$\frac{1}{(-2)^2+3(-2)-18}-\frac{3}{-2+6}\stackrel{?}{=}\frac{4}{-2-3}$$
$$-\frac{1}{20}-\frac{3}{4}\stackrel{?}{=}-\frac{4}{5}$$
$$\frac{-1-15}{20}\stackrel{?}{=}-\frac{4}{5}$$
$$\frac{-16}{20}\stackrel{?}{=}-\frac{4}{5}$$
$$-\frac{4}{5}=-\frac{4}{5}~True$$
Thus, the solution is $x=-2$.
