We are asked to solve the equation:

$\tan(x) = 2x \quad \text{in the interval} \quad (-\pi, 3\pi).$

Steps to Approach:

1. The function $\tan(x)$ is periodic with a period of $\pi$, while the linear term $2x$ is not periodic.
2. To find the solutions, we need to plot both $\tan(x)$ and $2x$ in the given interval and observe where these curves intersect.
3. We will analyze the behavior of $\tan(x)$, which has vertical asymptotes at $x = \frac{(2n+1)\pi}{2}$ for any integer $n$, and the linear function $2x$, which is continuous.

Let me plot the functions $\tan(x)$ and $2x$ in the range $(- \pi, 3\pi)$ to visualize the solutions.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to explain the solution approach in more detail without the graph, or would you like to try again later when the graphing tool is available?

Here are some related questions:

1. What are the asymptotes of the $\tan(x)$ function, and how do they affect the solution?
2. How many solutions would you expect between the interval $(- \pi, 3 \pi)$?
3. How can you solve equations involving $\tan(x)$ and a linear function algebraically?
4. Why do functions like $\tan(x)$ have periodic asymptotes?
5. Can you estimate the approximate x-values where the solutions occur?

Tip: When solving equations graphically, the points of intersection between the two functions represent the solutions.