If the complete set of value of x satisfying `|x-1|+|x-2|+|x-3|>=6` is `(-oo,a]uu[b,oo)`, then `a+b=`

Options

A. 2
B. 3
C. 6
D. 4

Answer

To solve the inequality \( |x-1| + |x-2| + |x-3| \geq 6 \), we will analyze the expression by considering different intervals based on the critical points \( x = 1, 2, 3 \).

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

### Step 2: Analyze the interval \( (-\infty, 1) \)
In this interval, all terms in the absolute values are negative:
\[ |x-1| = -(x-1) = -x + 1, \quad |x-2| = -(x-2) = -x + 2, \quad |x-3| = -(x-3) = -x + 3 \]
Thus, the inequality becomes:
\[ (-x + 1) + (-x + 2) + (-x + 3) \geq 6 \]
Simplifying this gives:
\[ -3x + 6 \geq 6 \implies -3x \geq 0 \implies x \leq 0 \]
Since \( x < 1 \) in this interval, we accept \( x \leq 0 \).

### Step 3: Analyze the interval \( [1, 2) \)
In this interval, we have:
\[ |x-1| = x - 1, \quad |x-2| = -(x-2) = -x + 2, \quad |x-3| = -(x-3) = -x + 3 \]
The inequality becomes:
\[ (x - 1) + (-x + 2) + (-x + 3) \geq 6 \]
Simplifying this gives:
\[ -x + 4 \geq 6 \implies -x \geq 2 \implies x \leq -2 \]
However, this contradicts the interval \( [1, 2) \), so there are no solutions in this interval.

### Step 4: Analyze the interval \( [2, 3) \)
In this interval, we have:
\[ |x-1| = x - 1, \quad |x-2| = x - 2, \quad |x-3| = -(x-3) = -x + 3 \]
The inequality becomes:
\[ (x - 1) + (x - 2) + (-x + 3) \geq 6 \]
Simplifying this gives:
\[ x + 0 \geq 6 \implies x \geq 6 \]
This contradicts the interval \( [2, 3) \), so there are no solutions here as well.

### Step 5: Analyze the interval \( [3, \infty) \)
In this interval, all terms in the absolute values are positive:
\[ |x-1| = x - 1, \quad |x-2| = x - 2, \quad |x-3| = x - 3 \]
The inequality becomes:
\[ (x - 1) + (x - 2) + (x - 3) \geq 6 \]
Simplifying this gives:
\[ 3x - 6 \geq 6 \implies 3x \geq 12 \implies x \geq 4 \]
This is valid in the interval \( [3, \infty) \).

### Step 6: Combine the results
From the analysis, we have:
- From \( (-\infty, 1) \): \( x \leq 0 \)
- From \( [1, 2) \): No solutions
- From \( [2, 3) \): No solutions
- From \( [3, \infty) \): \( x \geq 4 \)

Thus, the complete set of values satisfying the inequality is:
\[ (-\infty, 0] \cup [4, \infty) \]
This corresponds to \( a = 0 \) and \( b = 4 \).

### Final Calculation
Now, we need to find \( a + b \):
\[ a + b = 0 + 4 = 4 \]