Solve : `| x - 1| + |x - 2| + | x - 3 | gt 6 `

Options

A. `x in (-oo, 1)`
B. `x in (-oo, 0)`
C. `x in (4, oo)`
D. ` (2, oo)`

Answer

To solve the inequality \( |x - 1| + |x - 2| + |x - 3| > 6 \), we will consider different cases based on the values of \( x \) that affect the absolute values.

### Step 1: Identify the critical points
The critical points from the absolute values are \( x = 1 \), \( x = 2 \), and \( x = 3 \). These points will divide the number line into intervals.

### Step 2: Define the intervals
The intervals based on the critical points are:
1. \( x < 1 \)
2. \( 1 \leq x < 2 \)
3. \( 2 \leq x < 3 \)
4. \( x \geq 3 \)

### Step 3: Solve for each interval

#### Case 1: \( x \geq 3 \)
In this case, all the expressions inside the absolute values are positive:
\[ |x - 1| = x - 1, \quad |x - 2| = x - 2, \quad |x - 3| = x - 3 \]
So, the inequality becomes:
\[ (x - 1) + (x - 2) + (x - 3) > 6 \]
This simplifies to:
\[ 3x - 6 > 6 \]
\[ 3x > 12 \implies x > 4 \]

#### Case 2: \( 2 \leq x < 3 \)
Here, \( |x - 1| = x - 1 \), \( |x - 2| = x - 2 \), and \( |x - 3| = -(x - 3) = 3 - x \):
\[ (x - 1) + (x - 2) + (3 - x) > 6 \]
This simplifies to:
\[ x - 1 > 6 \implies x > 7 \]
However, this contradicts the case condition \( 2 \leq x < 3 \). Thus, there is no solution in this interval.

#### Case 3: \( 1 \leq x < 2 \)
In this case, \( |x - 1| = x - 1 \), \( |x - 2| = -(x - 2) = 2 - x \), and \( |x - 3| = -(x - 3) = 3 - x \):
\[ (x - 1) + (2 - x) + (3 - x) > 6 \]
This simplifies to:
\[ 4 - x > 6 \implies -x > 2 \implies x < -2 \]
Again, this contradicts the case condition \( 1 \leq x < 2 \). Thus, there is no solution in this interval.

#### Case 4: \( x < 1 \)
In this case, all expressions inside the absolute values are negative:
\[ |x - 1| = -(x - 1) = 1 - x, \quad |x - 2| = -(x - 2) = 2 - x, \quad |x - 3| = -(x - 3) = 3 - x \]
So, the inequality becomes:
\[ (1 - x) + (2 - x) + (3 - x) > 6 \]
This simplifies to:
\[ 6 - 3x > 6 \implies -3x > 0 \implies x < 0 \]

### Step 4: Combine the results
From the cases, we have:
- From Case 1: \( x > 4 \)
- From Case 2: No solution
- From Case 3: No solution
- From Case 4: \( x < 0 \)

Thus, the solution to the inequality \( |x - 1| + |x - 2| + |x - 3| > 6 \) is:
\[ x < 0 \quad \text{or} \quad x > 4 \]

### Final Answer
The solution set is \( (-\infty, 0) \cup (4, \infty) \).