Prove that the equation `2 sin x=|x|+a` has no solution for `a in ((3sqrt(3)-pi)/3, oo)`.

Explanation

To prove that the equation \( 2 \sin x = |x| + a \) has no solution for \( a \in \left( \frac{3\sqrt{3} - \pi}{3}, \infty \right) \), we will analyze the behavior of the two functions involved: \( y = 2 \sin x \) and \( y = |x| + a \).

### Step-by-Step Solution:

1. **Understanding the Functions**:
- The function \( y = 2 \sin x \) oscillates between -2 and 2 for all values of \( x \).
- The function \( y = |x| + a \) is a straight line that has a slope of 1. It starts at \( y = a \) when \( x = 0 \) and increases indefinitely as \( |x| \) increases.

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