Answer
$x = -6 \text{ or } x = 3$
Work Step by Step
Multiply each side of the equation by the LCD, which is $3x^2$ to eliminate the fractions:
$$\begin{align*}
3x^2\left(\frac{1}{x}+\frac{1}{3}\right)&=\frac{6}{x^2} \cdot 3x^2\\
3x^2\left(\frac{1}{x}\right)+3x^2\left(\frac{1}{3}\right)&=\frac{6}{x^2} \cdot 3x^2\\
\end{align*}$$
Simplify:
$$\require{cancel}
\begin{align*}
3x^\cancel{2}\left(\frac{1}{\cancel{x}}\right)+\cancel{3}x^2\left(\frac{1}{\cancel{3}}\right)&=\frac{6}{\cancel{x^2}} \cdot 3\cancel{x^2}\\
3x+x^2&=18\\
x^2+3x-18&=0\end{align*}$$
Factor the trinomial:
$$(x+6)(x-3)=0$$
Solve the equation using the Zero-Product Property by equating each factor to $0$, then solving each equation:
First factor:
$$\begin{align*}
x + 6 &= 0\\
x&=-6
\end{align*}$$
Second factor:
$$\begin{align*}
x -3 &= 0\\
x&=3
\end{align*}$$
To check the solution, plug in the values we just found for $x$ into the original equation:
First solution:
$$\begin{align*}
\frac{1}{-6} + \frac{1}{3} &\stackrel{?}{=} \frac{6}{(-6)^2}\\
-\frac{1}{6} + \frac{1}{3} &\stackrel{?}{=} \frac{6}{36}\\
-\frac{1}{6} + \frac{2}{6} &\stackrel{?}{=} \frac{1}{6}\\
\frac{1}{6}&\stackrel{\checkmark}{=} \frac{1}{6}\\
\end{align*}$$
Second solution:
$$\begin{align*}
\frac{1}{3} + \frac{1}{3} &\stackrel{?}{=} \frac{6}{3^2}\\
\frac{2}{3} &\stackrel{?}{=} \frac{6}{9}\\
\frac{2}{3}&\stackrel{\checkmark}{=} \frac{2}{3}\\
\end{align*}$$
