Solve \( 6 \tan x-7=0 \) for \( \pi \leq x \leq 2 \pi \) Give your answer correct to three significant figures.

To solve the equation \( 6 \tan x - 7 = 0 \), we first isolate \( \tan x \): \[ 6 \tan x = 7 \quad \Rightarrow \quad \tan x = \frac{7}{6} \] Next, we need to find the angles \( x \) within the interval \( \pi \leq x \leq 2\pi \). The tangent function is positive in the third and first quadrants; however, since we are concerned with the interval from \( \pi \) to \( 2\pi \), we focus on the third quadrant where the angle can be expressed as: \[ x = \pi + \tan^{-1}\left( \frac{7}{6} \right) \] Now, we need to calculate \( \tan^{-1}\left( \frac{7}{6} \right) \): Using a calculator, \[ \tan^{-1}\left( \frac{7}{6} \right) \approx 0.6947 \text{ radians} \] Thus, we find \( x \): \[ x \approx \pi + 0.6947 \] Calculating this gives: \[ x \approx 3.1416 + 0.6947 \approx 3.8363 \] Now, to ensure we round to three significant figures, we have: \[ x \approx 3.84 \] Now, we should also check for any other solutions in the interval. The tangent function has a period of \( \pi \), so the next solution would be: \[ x \approx 3.8363 + \pi \approx 3.8363 + 3.1416 = 6.978 \] However, this value \( 6.978 \) exceeds \( 2\pi \) (approximately \( 6.2832 \)), so it is not valid. Thus, the final answer is: \[ \boxed{3.84} \]