\( 1 \leftarrow \quad \) Solve the equation for \( x \), where \( x \) is restricted to the given interval. \[ y=8 \sec 3 x \text {, for } x \text { in }\left[0, \frac{\pi}{6}\right) \cup\left(\frac{\pi}{6}, \frac{\pi}{3}\right] \]

To solve the equation \( y = 8 \sec 3x \) for \( x \), we first set \( y = 1 \): \[ 1 = 8 \sec 3x \] This results in: \[ \sec 3x = \frac{1}{8} \] Given that \( \sec \theta = \frac{1}{\cos \theta} \), we rewrite it as: \[ \cos 3x = 8 \] However, the values of the cosine function are restricted between -1 and 1. Therefore, \( \cos 3x = 8 \) has no solution, leading us to conclude that there are no \( x \) values that satisfy the equation within the specified intervals. In conclusion, since the cosine function cannot equal 8, the solution set for the given equation in the intervals \( \left[0, \frac{\pi}{6}\right) \cup\left(\frac{\pi}{6}, \frac{\pi}{3}\right] \) is empty.