If [x] denotes greatest integer function less than or equal to x and {x} denotes fractional part then the number of real values of x satisfying equation (x−2)[x]={x}−1, is

Options

A. a. `-1`
B. b. 0
C. c. 1
D. d. none of these

Answer

To solve the equation \((x - 2)[x] = \{x\} - 1\), where \([x]\) denotes the greatest integer function (floor function) and \(\{x\}\) denotes the fractional part of \(x\), we can follow these steps:

### Step 1: Understand the Definitions
We know that:
- The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\).
- The fractional part \(\{x\}\) is defined as \(\{x\} = x - [x]\).

### Step 2: Rewrite the Equation
Substituting the definition of the fractional part into the equation, we have:
\[ (x - 2)[x] = (x - [x]) - 1 \]

### Step 3: Simplify the Equation
Rearranging the equation gives:
\[ (x - 2)[x] = x - [x] - 1 \]
This can be rewritten as:
\[ (x - 2)[x] + [x] + 1 = x \]
Combining terms yields:
\[ (x - 1)[x] + 1 = x \]

### Step 4: Isolate Terms
Now, isolate \(x\):
\[ (x - 1)[x] = x - 1 \]
If \(x - 1 \neq 0\) (i.e., \(x \neq 1\)), we can divide both sides by \(x - 1\):
\[ [x] = 1 \]

### Step 5: Analyze the Case \([x] = 1\)
The condition \([x] = 1\) implies:
\[ 1 \leq x < 2 \]
In this interval, the fractional part \(\{x\} = x - 1\).

### Step 6: Substitute Back to Check Validity
Substituting \([x] = 1\) into the original equation:
\[ (x - 2)(1) = (x - 1) - 1 \]
This simplifies to:
\[ x - 2 = x - 2 \]
This is true for all \(x\) in the interval \(1 \leq x < 2\).

### Step 7: Count the Real Values
The interval \(1 \leq x < 2\) contains infinitely many real numbers. Thus, the number of real values of \(x\) satisfying the equation is infinite.

### Conclusion
The number of real values of \(x\) satisfying the equation \((x - 2)[x] = \{x\} - 1\) is infinite.

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