To solve the inequality \( \tan x + 2 \sin x > 3x \), let's analyze both sides and examine where the inequality holds true.

### Steps for Solving
1. **Domain Consideration**: 
   - The functions \( \tan x \) and \( \sin x \) are periodic and have certain restrictions on their domains. Specifically, \( \tan x \) is undefined at \( x = \frac{\pi}{2} + n\pi \) for integer \( n \). So, we’ll need to consider intervals between these points where \( \tan x \) is defined.

2. **Compare Both Sides**:
   - Since this is an inequality involving \( \tan x \), \( \sin x \), and \( x \), it’s challenging to find an exact algebraic solution. However, you can analyze the inequality graphically or numerically by evaluating where \( \tan x + 2 \sin x \) is greater than \( 3x \) within the domain of \( x \).

3. **Graphical Approach**:
   - Graph the functions \( f(x) = \tan x + 2 \sin x \) and \( g(x) = 3x \).
   - Look for intervals where \( f(x) > g(x) \), which would give the solutions to the inequality. 

4. **Approximate Values**:
   - If needed, numerical methods or specific values can be used to approximate where \( f(x) > g(x) \).

Would you like a detailed solution using a graphical or numerical approach to find approximate intervals? 

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**Related Questions:**

1. How does the periodicity of trigonometric functions affect inequalities?
2. What methods exist for solving inequalities involving trigonometric and linear terms?
3. Can transformations help simplify inequalities like \( \tan x + 2 \sin x > 3x \)?
4. How can the properties of the sine and tangent functions be used in inequality proofs?
5. What are typical domains for different trigonometric functions?

**Tip:** When working with trigonometric inequalities, analyzing domains and using graphing can be especially helpful for visualizing solutions.

Discover methods for solving the trigonometric inequality tan(x) + 2sin(x) > 3x. This problem explores periodicity, graphing techniques, and comparison of trigonometric and linear functions. Ideal for Grades 10-12 students wanting to deepen their understanding of inequality solving.

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