Solve the equation `4[x]=x+{x}`

Answer

To solve the equation \( 4[x] = x + \{x\} \), we will follow these steps:

### Step 1: Understand the components of the equation
The equation involves two parts: the greatest integer function \([x]\) (also known as the floor function) and the fractional part \(\{x\}\). We know that:
\[ x = [x] + \{x\} \]
where \([x]\) is the greatest integer less than or equal to \(x\) and \(\{x\} = x - [x]\).

### Step 2: Substitute the expression for \(x\)
Using the relationship between \(x\), \([x]\), and \(\{x\}\), we can rewrite the equation:
\[ 4[x] = [x] + \{x\} \]
Substituting \(\{x\} = x - [x]\) into the equation gives:
\[ 4[x] = [x] + (x - [x]) \]
This simplifies to:
\[ 4[x] = x \]

### Step 3: Express \(x\) in terms of \([x]\)
From the equation \(4[x] = x\), we can express \(x\) as:
\[ x = 4[x] \]

### Step 4: Analyze the fractional part
Since \([x]\) is an integer, we can denote \([x] = n\) where \(n\) is an integer. Thus, we have:
\[ x = 4n \]
Now, we can express \(x\) in terms of its integer and fractional parts:
\[ x = n + \{x\} \]
Substituting for \(x\):
\[ 4n = n + \{x\} \]
This leads to:
\[ \{x\} = 4n - n = 3n \]

### Step 5: Determine the range of \(\{x\}\)
The fractional part \(\{x\}\) must satisfy:
\[ 0 \leq \{x\} < 1 \]
Substituting \(3n\) for \(\{x\}\):
\[ 0 \leq 3n < 1 \]
This implies:
\[ 0 \leq n < \frac{1}{3} \]
Since \(n\) is an integer, the only possible value for \(n\) is \(0\).

### Step 6: Find the corresponding \(x\)
If \(n = 0\), then:
\[ [x] = 0 \quad \text{and} \quad x = 4 \cdot 0 = 0 \]
Thus, \(x\) can take values in the range:
\[ 0 \leq x < 1 \]

### Conclusion
The solution to the equation \(4[x] = x + \{x\}\) is:
\[ x \in [0, 1) \]