If x satisfies `|x-1| + |x-2|+|x-3|gt6,` then :
i)x ∈ (−∞,1)
ii)x ∈ (−∞,0)
iii)x ∈ (4,∞)
iv) x ∈ (2,∞)

Options

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

Answer

To solve the inequality \( |x-1| + |x-2| + |x-3| > 6 \), we will analyze the expression by considering different intervals based on the critical points where the absolute values change, which are \( x = 1 \), \( x = 2 \), and \( x = 3 \).

### Step 1: Identify the intervals
The critical points divide the number line into four intervals:
1. \( (-\infty, 1) \)
2. \( [1, 2) \)
3. \( [2, 3) \)
4. \( [3, \infty) \)

### Step 2: Analyze each interval

**Interval 1: \( x < 1 \)**
- Here, \( |x-1| = 1-x \), \( |x-2| = 2-x \), \( |x-3| = 3-x \).
- Therefore, the inequality becomes:
\[ (1-x) + (2-x) + (3-x) > 6 \]
Simplifying this:
\[ 6 - 3x > 6 \implies -3x > 0 \implies x < 0 \]
- Thus, for this interval, we have \( x \in (-\infty, 0) \).

**Interval 2: \( 1 \leq x < 2 \)**
- Here, \( |x-1| = x-1 \), \( |x-2| = 2-x \), \( |x-3| = 3-x \).
- The inequality becomes:
\[ (x-1) + (2-x) + (3-x) > 6 \]
Simplifying this:
\[ 4 - x > 6 \implies -x > 2 \implies x < -2 \]
- However, this is not possible in the interval \( [1, 2) \). Thus, there are no solutions in this interval.

**Interval 3: \( 2 \leq x < 3 \)**
- Here, \( |x-1| = x-1 \), \( |x-2| = x-2 \), \( |x-3| = 3-x \).
- The inequality becomes:
\[ (x-1) + (x-2) + (3-x) > 6 \]
Simplifying this:
\[ x > 6 \implies \text{no solutions in this interval.} \]

**Interval 4: \( x \geq 3 \)**
- Here, \( |x-1| = x-1 \), \( |x-2| = x-2 \), \( |x-3| = x-3 \).
- The inequality becomes:
\[ (x-1) + (x-2) + (x-3) > 6 \]
Simplifying this:
\[ 3x - 6 > 6 \implies 3x > 12 \implies x > 4 \]
- Thus, for this interval, we have \( x \in (4, \infty) \).

### Step 3: Combine the results
From our analysis, we found:
- From interval 1: \( x \in (-\infty, 0) \)
- From interval 4: \( x \in (4, \infty) \)

Therefore, the solution to the inequality \( |x-1| + |x-2| + |x-3| > 6 \) is:
\[ x \in (-\infty, 0) \cup (4, \infty) \]

### Conclusion
Among the options provided, the correct answer is:
- iii) \( x \in (4, \infty) \)