The number of roots of the equation, `x-2/(x-1)=1-2/(x-1)` is 0 (b) 1 (c) 2 (d) 3

Options

A. 1
B. 2
C. 0
D. infinitely many

Answer

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

### Step 1: Simplify the Equation
We start with the equation:
\[ \frac{x - 2}{x - 1} = 1 - \frac{2}{x - 1} \]

### Step 2: Combine the Right Side
To combine the right side, we can rewrite \(1\) as \(\frac{x - 1}{x - 1}\):
\[ 1 - \frac{2}{x - 1} = \frac{x - 1}{x - 1} - \frac{2}{x - 1} = \frac{x - 1 - 2}{x - 1} = \frac{x - 3}{x - 1} \]

### Step 3: Set the Equation Equal
Now we have:
\[ \frac{x - 2}{x - 1} = \frac{x - 3}{x - 1} \]

### Step 4: Cross Multiply
Since the denominators are the same (assuming \(x \neq 1\)), we can cross-multiply:
\[ (x - 2)(x - 1) = (x - 3)(x - 1) \]

### Step 5: Expand Both Sides
Expanding both sides gives:
\[ x^2 - 3x + 2 = x^2 - 4x + 3 \]

### Step 6: Simplify the Equation
Now, we can simplify by subtracting \(x^2\) from both sides:
\[ -3x + 2 = -4x + 3 \]

### Step 7: Rearrange the Equation
Rearranging gives:
\[ -3x + 4x = 3 - 2 \]
\[ x = 1 \]

### Step 8: Check for Validity
Now we need to check if \(x = 1\) is a valid solution by substituting it back into the original equation:
\[ \frac{1 - 2}{1 - 1} = 1 - \frac{2}{1 - 1} \]
Both sides result in \(\frac{-1}{0}\) which is undefined.

### Conclusion
Since \(x = 1\) does not provide a valid solution, the equation has no roots.

Thus, the number of roots of the equation is **0** (option C).

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