The number of solutions of `|[x]-2x|=4, "where" [x]]` is the greatest integer less than or equal to x, is

Options

A. 2
B. 4
C. 1
D. infinite

Explanation

To solve the equation \( |[x] - 2x| = 4 \), where \([x]\) is the greatest integer less than or equal to \(x\), we will analyze the problem step by step.

### Step 1: Define the greatest integer function
The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\). We can denote \([x] = n\), where \(n\) is an integer. Thus, we can express \(x\) as:
\[ x = n + \lambda \]
where \(0 \leq \lambda < 1\) is the fractional part of \(x\).

### Step 2: Rewrite the equation
Substituting \([x] = n\) into the equation, we have:
\[ |n - 2(n + \lambda)| = 4 \]
This simplifies to:
\[ |n - 2n - 2\lambda| = 4 \quad \Rightarrow \quad |-n - 2\lambda| = 4 \]

### Step 3: Remove the absolute value
The equation \( |-n - 2\lambda| = 4 \) gives us two cases to consider:
1. \(-n - 2\lambda = 4\)
2. \(-n - 2\lambda = -4\)

### Step 4: Solve the first case
For the first case:
\[ -n - 2\lambda = 4 \quad \Rightarrow \quad n + 2\lambda = -4 \quad \Rightarrow \quad n = -4 - 2\lambda \]

### Step 5: Solve the second case
For the second case:
\[ -n - 2\lambda = -4 \quad \Rightarrow \quad n + 2\lambda = 4 \quad \Rightarrow \quad n = 4 - 2\lambda \]

### Step 6: Analyze the cases
Now we have two expressions for \(n\):
1. \(n = -4 - 2\lambda\)
2. \(n = 4 - 2\lambda\)

#### Case 1: \(n = -4 - 2\lambda\)
Since \(n\) is an integer, \(2\lambda\) must be an integer. The only value of \(\lambda\) that satisfies \(0 \leq \lambda < 1\) and makes \(2\lambda\) an integer is \(\lambda = \frac{1}{2}\). Thus:
\[ n = -4 - 2 \cdot \frac{1}{2} = -4 - 1 = -5 \]
This gives:
\[ x = n + \lambda = -5 + \frac{1}{2} = -\frac{9}{2} \]

#### Case 2: \(n = 4 - 2\lambda\)
Again, since \(n\) is an integer, \(2\lambda\) must be an integer. The only value of \(\lambda\) that satisfies \(0 \leq \lambda < 1\) and makes \(2\lambda\) an integer is \(\lambda = \frac{1}{2}\). Thus:
\[ n = 4 - 2 \cdot \frac{1}{2} = 4 - 1 = 3 \]
This gives:
\[ x = n + \lambda = 3 + \frac{1}{2} = \frac{7}{2} \]

### Step 7: Consider integer solutions
Now we also need to consider when \(x\) is an integer. In this case, we have:
\[ |x - 2x| = | -x | = 4 \quad \Rightarrow \quad x = 4 \quad \text{or} \quad x = -4 \]

### Step 8: Collect all solutions
The solutions we have found are:
1. \(x = -\frac{9}{2}\)
2. \(x = \frac{7}{2}\)
3. \(x = 4\)
4. \(x = -4\)

### Conclusion
Thus, the total number of solutions to the equation \( |[x] - 2x| = 4 \) is 4.