The solution set of the inequation ` (x-1)/(x-2) gt 2,` is

Options

A. (2, 3)
B. [2, 3]
C. `(-oo, 2) cup (3, oo)`
D. none of these

Explanation

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

### Step 1: Rearranging the Inequality
We start by moving all terms to one side of the inequality:
\[ \frac{x-1}{x-2} - 2 > 0 \]

### Step 2: Finding a Common Denominator
Next, we need to combine the terms on the left side. The common denominator is \(x - 2\):
\[ \frac{x-1 - 2(x-2)}{x-2} > 0 \]

### Step 3: Simplifying the Numerator
Now, simplify the numerator:
\[ x - 1 - 2x + 4 = -x + 3 \]
Thus, we have:
\[ \frac{-x + 3}{x - 2} > 0 \]

### Step 4: Factoring Out the Negative Sign
We can factor out the negative sign from the numerator:
\[ \frac{-(x - 3)}{x - 2} > 0 \]
This can be rewritten as:
\[ \frac{x - 3}{x - 2} < 0 \]

### Step 5: Finding Critical Points
The critical points occur where the numerator and denominator are zero:
- \(x - 3 = 0 \Rightarrow x = 3\)
- \(x - 2 = 0 \Rightarrow x = 2\)

### Step 6: Testing Intervals
We will test the intervals determined by the critical points \(x = 2\) and \(x = 3\):
1. Choose a test point in the interval \((-∞, 2)\), e.g., \(x = 0\):
\[ \frac{0 - 3}{0 - 2} = \frac{-3}{-2} > 0 \quad \text{(not a solution)} \]
2. Choose a test point in the interval \((2, 3)\), e.g., \(x = 2.5\):
\[ \frac{2.5 - 3}{2.5 - 2} = \frac{-0.5}{0.5} < 0 \quad \text{(solution)} \]
3. Choose a test point in the interval \((3, ∞)\), e.g., \(x = 4\):
\[ \frac{4 - 3}{4 - 2} = \frac{1}{2} > 0 \quad \text{(not a solution)} \]

### Step 7: Writing the Solution Set
The solution set is the interval where the inequality holds true:
\[ x \in (2, 3) \]

### Final Answer
Thus, the solution set of the inequation \(\frac{x-1}{x-2} > 2\) is:
\[ \boxed{(2, 3)} \]