To estimate the solutions of the equation $\sin(x) = 0.1$ graphically, we can find where the graph of $y = \sin(x)$ intersects the line $y = 0.1$ in the interval $-3\pi < x < 3\pi$.

The sine function oscillates between -1 and 1, so we will look for the points where $y = 0.1$ intersects with the sine curve.

Steps to estimate solutions:

1. Identify the points of intersection: You need to look at the graph to determine where the line $y = 0.1$ crosses the sine curve.
2. Determine the approximate x-values: In the interval $-3\pi$ to $3\pi$, you can find several intersections.

Using this method, the solutions (estimated from the graph) are approximately:

1. $x \approx -3.0416$
2. $x \approx -2.3020$
3. $x \approx -0.1000$
4. $x \approx 0.1000$
5. $x \approx 2.3020$
6. $x \approx 3.0416$

Thus, rounding these values to four decimal places, the solutions are:

$x \approx -3.0416, -2.3020, -0.1000, 0.1000, 2.3020, 3.0416$

Final answer: $-3.0416, -2.3020, -0.1000, 0.1000, 2.3020, 3.0416$

Would you like more details or have any questions? Here are some related questions:

1. How does the graph of $y = \sin(x)$ behave over different intervals?
2. What is the general solution for $\sin(x) = k$?
3. Can you explain how to derive these solutions algebraically?
4. How would you find intersections for different values of $k$?
5. What are the properties of the sine function that affect its intersections with horizontal lines?

Tip: Always check the periodicity of the sine function to identify additional solutions beyond the given interval.