`x gt `sin x for

Options

A. all `x gt 0`
B. all `x lt 0`
C. all x for which `x ne 0`
D. all x

Explanation

To solve the inequality \( x > \sin x \), we can analyze the behavior of the functions involved. Here’s a step-by-step solution:

### Step 1: Understand the Functions
We know that:
- The function \( \sin x \) oscillates between -1 and 1 for all real numbers \( x \).
- The function \( y = x \) is a straight line with a slope of 1 that passes through the origin.

### Step 2: Analyze the Intersection Points
To find where \( x = \sin x \), we can look for points of intersection between the line \( y = x \) and the curve \( y = \sin x \).

### Step 3: Check at Key Points
- At \( x = 0 \):
\[ \sin(0) = 0 \quad \Rightarrow \quad 0 = 0 \]
So, \( x = 0 \) is a point of intersection.

- For \( x > 0 \):
- As \( x \) increases, \( \sin x \) will always be less than \( x \) because \( \sin x \) reaches a maximum of 1, while \( x \) continues to increase without bound.

- For \( x < 0 \):
- As \( x \) decreases, \( \sin x \) will oscillate between -1 and 1, but since \( x \) is negative and decreasing, \( x \) will be less than \( \sin x \) for all negative values of \( x \).

### Step 4: Conclusion
From our analysis:
- For \( x > 0 \), \( x > \sin x \).
- For \( x < 0 \), \( x < \sin x \).
- At \( x = 0 \), they are equal.

Thus, the inequality \( x > \sin x \) holds true for all \( x > 0 \).

### Final Answer
The correct option is: **All \( x > 0 \)**.

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