Solve `sqrt(x-1)/(x-2) lt 0`

Explanation

To solve the inequality \(\frac{\sqrt{x-1}}{x-2} < 0\), we will follow these steps:

### Step 1: Determine the domain of the expression
The expression \(\sqrt{x-1}\) is defined only when the argument is non-negative. Therefore, we need:
\[ x - 1 \geq 0 \implies x \geq 1 \]

### Step 2: Analyze the inequality
Next, we need to analyze the inequality \(\frac{\sqrt{x-1}}{x-2} < 0\). This fraction will be negative if the numerator and denominator have opposite signs.

- The numerator \(\sqrt{x-1}\) is non-negative (it is zero or positive) for \(x \geq 1\).
- The denominator \(x - 2\) will be negative when \(x < 2\).

### Step 3: Combine the conditions
From the analysis:
1. The numerator \(\sqrt{x-1} \geq 0\) for \(x \geq 1\).
2. The denominator \(x - 2 < 0\) for \(x < 2\).

For the fraction to be negative, we need:
- The numerator to be positive (which is true for \(x > 1\)).
- The denominator to be negative (which is true for \(x < 2\)).

Thus, we combine these conditions:
\[ 1 < x < 2 \]

### Step 4: Write the final solution
The solution to the inequality \(\frac{\sqrt{x-1}}{x-2} < 0\) is:
\[ 1 < x < 2 \]

### Summary of the solution:
The final answer is:
\[ x \in (1, 2) \]