We can solve this using trigonometric identities.
We know that $\tan x = \frac{\sin x}{\cos x}$ and $\cot x = \frac{\cos x}{\sin x}$.
So, $\tan x + \cot x = \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x}$.
To add these fractions, find a common denominator, which is $\sin x \cos x$.
$= \frac{\sin^2 x + \cos^2 x}{\sin x \cos x}$.
Using the Pythagorean identity $\sin^2 x + \cos^2 x = 1$, the expression simplifies to:
$= \frac{1}{\sin x \cos x}$.
We are given $\sin x = \frac{3}{5}$.
We need to find $\cos x$. Using the Pythagorean identity again: $\cos x = \sqrt{1 - \sin^2 x}$.
$\cos x = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \sqrt{1 - \frac{9}{25}} = \sqrt{\frac{25-9}{25}} = \sqrt{\frac{16}{25}} = \frac{4}{5}$.
Now substitute the values of $\sin x$ and $\cos x$ into our simplified expression:
$\tan x + \cot x = \frac{1}{\left(\frac{3}{5}\right) \times \left(\frac{4}{5}\right)} = \frac{1}{\frac{12}{25}}$.
$= \frac{25}{12}$.