The equation shown in the image is:

$-2\sin(x) = 5\sin(x) - \frac{7}{2}$

Step-by-Step Solution:

1. Move all terms involving $\sin(x)$ to one side: $-2\sin(x) - 5\sin(x) = -\frac{7}{2}$ This simplifies to: $-7\sin(x) = -\frac{7}{2}$

2. Solve for $\sin(x)$: Divide both sides by $-7$: $\sin(x) = \frac{1}{2}$

3. Find the general solution for $\sin(x) = \frac{1}{2}$: The sine function equals $\frac{1}{2}$ at the angles: $x = \frac{\pi}{6} + 2k\pi \quad \text{or} \quad x = \pi - \frac{\pi}{6} + 2k\pi$ This simplifies to: $x = \frac{\pi}{6} + 2k\pi \quad \text{or} \quad x = \frac{5\pi}{6} + 2k\pi$ where $k$ is an integer.

4. Answer format: The solutions are presented in terms of $(-1)^n$, where $n \in \mathbb{Z}$ represents the periodic nature of the sine function. Looking at the answer choices in the image, the correct general solution is:

   $x = (-1)^n \frac{\pi}{6} + \pi n, \, n \in \mathbb{Z}$

Thus, the correct option is the last one.

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. How can we generalize solutions to other trigonometric equations?
2. What is the geometric interpretation of $\sin(x) = \frac{1}{2}$?
3. How do we solve trigonometric equations involving multiple angles?
4. What are the differences in solving for sine, cosine, and tangent equations?
5. How do phase shifts affect the solutions of trigonometric equations?

Tip: When solving trigonometric equations, always consider the periodicity and symmetry of the trigonometric functions to find all possible solutions.