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 graphs of the two functions involved: \( y = 2 \sin x \) and \( y = |x| + a \).

### Step 1: Understand the functions
1. The function \( y = 2 \sin x \) oscillates between -2 and 2, since the sine function oscillates between -1 and 1. Therefore, the range of \( y = 2 \sin x \) is \([-2, 2]\).
2. The function \( y = |x| + a \) is a V-shaped graph that intersects the y-axis at \( a \) and has a slope of 1 for \( x \geq 0 \) and -1 for \( x < 0 \).