How to simplify trigonometric inequality? $| 3 ^ { \tan ( \pi x ) } - 3 ^ { 1 - \tan ( \pi x ) } | \geq 2$

Let $y=3^{\tan(\pi x)}$ so that $3^{1-\tan(\pi x)}=\frac{3^1}{3^{\tan(\pi x)}}=\frac{3}{y}$.  Now, your inequality becomes $|y-\frac{3}{y}|\ge 2$.
This can be solved with the boundary algorithm (sometimes called the test-point method)—solve the corresponding equation, $|y-\frac{3}{y}|=2$, plot those solutions on a number line, plot any values of $y$ for which part(s) of the equation are undefined (e.g. $y=0$), and test a value in each resulting interval to see if that interval satisfies the inequality you're solving.
Once you've got a solution for $y$, go back and use that to solve for $x$.
