How many values of [x] exists such that `4{x} = x + [x]`

Options

A. A) 2
B. B) 1
C. C) 0
D. D) None of these

Answer

To solve the equation \( 4\{x\} = x + [x] \), we will break it down step by step.

### Step 1: Understand the notation
The notation \( [x] \) represents the greatest integer less than or equal to \( x \) (the integer part), and \( \{x\} \) represents the fractional part of \( x \). The relationship can be expressed as:
\[ x = [x] + \{x\} \]

### Step 2: Rewrite the equation
Given the equation:
\[ 4\{x\} = x + [x] \]
Substituting \( x = [x] + \{x\} \) into the equation gives:
\[ 4\{x\} = ([x] + \{x\}) + [x] \]
This simplifies to:
\[ 4\{x\} = 2[x] + \{x\} \]

### Step 3: Rearrange the equation
Now, we can rearrange the equation:
\[ 4\{x\} - \{x\} = 2[x] \]
This simplifies to:
\[ 3\{x\} = 2[x] \]

### Step 4: Express the fractional part
From the equation \( 3\{x\} = 2[x] \), we can express the fractional part \( \{x\} \):
\[ \{x\} = \frac{2}{3}[x] \]

### Step 5: Determine the range of \( \{x\} \)
Since \( \{x\} \) is the fractional part, it must satisfy:
\[ 0 \leq \{x\} < 1 \]
Substituting \( \{x\} = \frac{2}{3}[x] \) into this inequality gives:
\[ 0 \leq \frac{2}{3}[x] < 1 \]

### Step 6: Solve the inequality
To solve the inequality \( \frac{2}{3}[x] < 1 \), we multiply both sides by \( \frac{3}{2} \):
\[ [x] < \frac{3}{2} \]
This means:
\[ [x] \leq 1 \]
Since \( [x] \) is an integer, the possible integer values for \( [x] \) are 0 and 1.

### Step 7: Count the values
Thus, the integer values \( [x] \) can take are:
- \( [x] = 0 \)
- \( [x] = 1 \)

### Conclusion
Therefore, there are **2 values** of \( [x] \) that satisfy the equation \( 4\{x\} = x + [x] \).