Solve the equation: \(\sqrt{9/6} = 8 - 5 \sqrt{x-1}\)

Solution:

We are asked to solve the equation \(\sqrt{9/6} = 8 - 5 \sqrt{x-1}\).

1. Simplify the left side: \(\sqrt{9/6} = \sqrt{3/2} = \frac{\sqrt{3}}{\sqrt{2}} = \frac{\sqrt{6}}{2}\).
2. Isolate the square root term: \(\frac{\sqrt{6}}{2} = 8 - 5 \sqrt{x-1}\)
   \(5 \sqrt{x-1} = 8 - \frac{\sqrt{6}}{2}\)
   \(5 \sqrt{x-1} = \frac{16 - \sqrt{6}}{2}\)
   \(\sqrt{x-1} = \frac{16 - \sqrt{6}}{10}\).
3. Square both sides: \(x-1 = \left(\frac{16 - \sqrt{6}}{10}\right)^2\)
   \(x-1 = \frac{16^2 - 2 * 16 * \sqrt{6} + (\sqrt{6})^2}{100}\)
   \(x-1 = \frac{256 - 32\sqrt{6} + 6}{100}\)
   \(x-1 = \frac{262 - 32\sqrt{6}}{100}\)
   \(x-1 = \frac{131 - 16\sqrt{6}}{50}\).
4. Solve for x: \(x = 1 + \frac{131 - 16\sqrt{6}}{50}\)
   \(x = \frac{50 + 131 - 16\sqrt{6}}{50}\)
   \(x = \frac{181 - 16\sqrt{6}}{50}\).

Answer: \(x = \frac{181 - 16\sqrt{6}}{50}\)