Consider the equation is real number x and a real parameter `lamda, |x-1| -|x-2| + |x-4| =lamda ` Then for `lamda ge 1,` the number of solutions, the equation can have is/are :

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the equation \( |x - 1| - |x - 2| + |x - 4| = \lambda \) for \( \lambda \geq 1 \), we will analyze the expression based on the critical points where the absolute value functions change, which are \( x = 1, 2, 4 \).

### Step 1: Identify intervals based on critical points
The critical points divide the number line into the following intervals:
1. \( x < 1 \)
2. \( 1 \leq x < 2 \)
3. \( 2 \leq x < 4 \)
4. \( x \geq 4 \)

### Step 2: Analyze each interval

**Interval 1: \( x < 1 \)**
- Here, \( |x - 1| = -(x - 1) = 1 - x \)
- \( |x - 2| = -(x - 2) = 2 - x \)
- \( |x - 4| = -(x - 4) = 4 - x \)

Substituting these into the equation:
\[ (1 - x) - (2 - x) + (4 - x) = \lambda \]
Simplifying:
\[ 1 - x - 2 + x + 4 - x = \lambda \implies 3 - x = \lambda \implies x = 3 - \lambda \]
For \( x < 1 \), we need \( 3 - \lambda < 1 \) which gives:
\[ \lambda > 2 \]

**Interval 2: \( 1 \leq x < 2 \)**
- Here, \( |x - 1| = x - 1 \)
- \( |x - 2| = 2 - x \)
- \( |x - 4| = 4 - x \)

Substituting these into the equation:
\[ (x - 1) - (2 - x) + (4 - x) = \lambda \]
Simplifying:
\[ x - 1 - 2 + x + 4 - x = \lambda \implies x + 1 = \lambda \implies x = \lambda - 1 \]
For \( 1 \leq x < 2 \), we need:
\[ 1 \leq \lambda - 1 < 2 \implies 2 \leq \lambda < 3 \]

**Interval 3: \( 2 \leq x < 4 \)**
- Here, \( |x - 1| = x - 1 \)
- \( |x - 2| = x - 2 \)
- \( |x - 4| = 4 - x \)

Substituting these into the equation:
\[ (x - 1) - (x - 2) + (4 - x) = \lambda \]
Simplifying:
\[ x - 1 - x + 2 + 4 - x = \lambda \implies 5 - x = \lambda \implies x = 5 - \lambda \]
For \( 2 \leq x < 4 \), we need:
\[ 2 \leq 5 - \lambda < 4 \implies 1 < \lambda \leq 3 \]

**Interval 4: \( x \geq 4 \)**
- Here, \( |x - 1| = x - 1 \)
- \( |x - 2| = x - 2 \)
- \( |x - 4| = x - 4 \)

Substituting these into the equation:
\[ (x - 1) - (x - 2) + (x - 4) = \lambda \]
Simplifying:
\[ x - 1 - x + 2 + x - 4 = \lambda \implies x - 3 = \lambda \implies x = \lambda + 3 \]
For \( x \geq 4 \), we need:
\[ \lambda + 3 \geq 4 \implies \lambda \geq 1 \]

### Step 3: Summary of solutions based on \( \lambda \)
- For \( \lambda > 2 \): 1 solution from interval 1.
- For \( 2 \leq \lambda < 3 \): 1 solution from interval 2.
- For \( 1 < \lambda \leq 3 \): 1 solution from interval 3.
- For \( \lambda = 3 \): 1 solution from interval 4.

### Conclusion
Thus, the number of solutions for \( \lambda \geq 1 \) can be:
- 1 solution when \( \lambda = 1 \)
- 2 solutions when \( 1 < \lambda < 2 \)
- 3 solutions when \( \lambda = 2 \)
- 4 solutions when \( 2 < \lambda < 3 \)
- 3 solutions when \( \lambda = 3 \)
- 2 solutions when \( \lambda > 3 \)

### Final Answer
The number of solutions can be \( 1, 2, 3, \) or \( 4 \).