The maximum number of values of x is `|x-2| + |x-4| = 2` is

Options

A. 1
B. 2
C. 3
D. Infinitely many

Answer

To solve the equation \( |x - 2| + |x - 4| = 2 \), we will analyze it by breaking it down into different cases based on the values of \( x \).

### Step 1: Identify the critical points
The critical points where the expressions inside the absolute values change are \( x = 2 \) and \( x = 4 \). We will consider three cases based on these points.

### Step 2: Case 1 - \( x < 2 \)
In this case, both \( |x - 2| \) and \( |x - 4| \) will be negative:
\[ |x - 2| = -(x - 2) = -x + 2 \]
\[ |x - 4| = -(x - 4) = -x + 4 \]
Substituting these into the equation:
\[ (-x + 2) + (-x + 4) = 2 \]
This simplifies to:
\[ -2x + 6 = 2 \]
Rearranging gives:
\[ -2x = 2 - 6 \]
\[ -2x = -4 \]
\[ x = 2 \]
Since \( x = 2 \) does not satisfy \( x < 2 \), there are no solutions in this case.

### Step 3: Case 2 - \( 2 \leq x < 4 \)
In this case, \( |x - 2| \) is positive and \( |x - 4| \) is negative:
\[ |x - 2| = x - 2 \]
\[ |x - 4| = -(x - 4) = -x + 4 \]
Substituting these into the equation:
\[ (x - 2) + (-x + 4) = 2 \]
This simplifies to:
\[ x - 2 - x + 4 = 2 \]
\[ 2 = 2 \]
This is always true, which means there are infinitely many solutions for \( x \) in the interval \( [2, 4) \).

### Step 4: Case 3 - \( x \geq 4 \)
In this case, both \( |x - 2| \) and \( |x - 4| \) are positive:
\[ |x - 2| = x - 2 \]
\[ |x - 4| = x - 4 \]
Substituting these into the equation:
\[ (x - 2) + (x - 4) = 2 \]
This simplifies to:
\[ 2x - 6 = 2 \]
Rearranging gives:
\[ 2x = 8 \]
\[ x = 4 \]
Since \( x = 4 \) satisfies \( x \geq 4 \), this is a valid solution.

### Conclusion
From the three cases, we find:
- Case 1: No solutions
- Case 2: Infinitely many solutions in the interval \( [2, 4) \)
- Case 3: One solution at \( x = 4 \)

Thus, the maximum number of values of \( x \) that satisfy the equation \( |x - 2| + |x - 4| = 2 \) is **infinitely many**.