The number of solutions of `|[x]-2x|=43`, where `[x]` denotes the greatest integer `lex` is

Answer

To solve the equation \(|[x] - 2x| = 43\), where \([x]\) denotes the greatest integer less than or equal to \(x\), we will analyze the equation step by step.

### Step 1: Understand the Equation
The equation \(|[x] - 2x| = 43\) can be interpreted in two cases based on the definition of absolute value.

### Step 2: Case 1 - \(x\) is an Integer
If \(x\) is an integer, then \([x] = x\). The equation simplifies to:
\[ |x - 2x| = | -x | = 43 \]
This gives us two equations:
1. \(-x = 43\) which leads to \(x = -43\)
2. \(-x = -43\) which leads to \(x = 43\)

Thus, from Case 1, we have two integer solutions: \(x = -43\) and \(x = 43\).

### Step 3: Case 2 - \(x\) is Not an Integer
If \(x\) is not an integer, we can express \(x\) as \(x = n + \lambda\), where \(n = [x]\) (an integer) and \(\lambda\) is the fractional part of \(x\) such that \(0 < \lambda < 1\).

Substituting into the equation gives:
\[ |n - 2(n + \lambda)| = |n - 2n - 2\lambda| = |-n - 2\lambda| = 43 \]
This leads to two scenarios:
1. \(-n - 2\lambda = 43\)
2. \(-n - 2\lambda = -43\)

### Step 4: Solve the First Scenario
From the first scenario:
\[ -n - 2\lambda = 43 \implies n + 2\lambda = -43 \]
This implies that \(n\) must be negative because \(n\) is an integer. Rearranging gives:
\[ 2\lambda = -43 - n \implies \lambda = \frac{-43 - n}{2} \]
Since \(0 < \lambda < 1\), we have:
\[ 0 < \frac{-43 - n}{2} < 1 \]
This leads to two inequalities:
1. \(-43 - n > 0 \implies n < -43\)
2. \(-43 - n < 2 \implies n > -45\)

Thus, \(n\) can take the values \(-44\) and \(-43\).

### Step 5: Solve the Second Scenario
From the second scenario:
\[ -n - 2\lambda = -43 \implies n + 2\lambda = 43 \]
Rearranging gives:
\[ 2\lambda = 43 - n \implies \lambda = \frac{43 - n}{2} \]
Again, since \(0 < \lambda < 1\), we have:
\[ 0 < \frac{43 - n}{2} < 1 \]
This leads to:
1. \(43 - n > 0 \implies n < 43\)
2. \(43 - n < 2 \implies n > 41\)

Thus, \(n\) can take the values \(42\) and \(41\).

### Step 6: Combine the Solutions
Now we have the following solutions from both cases:
- From Case 1: \(x = -43\) and \(x = 43\)
- From Case 2:
- For \(n = -44\):
\[ \lambda = \frac{-43 - (-44)}{2} = \frac{1}{2} \implies x = -44 + \frac{1}{2} = -43.5 \]
- For \(n = -43\):
\[ \lambda = \frac{-43 - (-43)}{2} = 0 \implies x = -43 + 0 = -43 \text{ (already counted)} \]
- For \(n = 42\):
\[ \lambda = \frac{43 - 42}{2} = \frac{1}{2} \implies x = 42 + \frac{1}{2} = 42.5 \]
- For \(n = 41\):
\[ \lambda = \frac{43 - 41}{2} = 1 \implies x = 41 + 1 = 42 \text{ (not valid since } \lambda < 1\text{)} \]

### Final Count of Solutions
Thus, the valid solutions are:
1. \(x = -43\)
2. \(x = 43\)
3. \(x = -43.5\)
4. \(x = 42.5\)

In total, we have **4 solutions**.