Let a,b,c,d be real numbers such that |a-b|=2, |b-c|=3, |c-d|=4 Then the sum of all possible values of |a-d|=

Options

A. `9`
B. `18`
C. `24`
D. `30`

Explanation

To solve the problem step by step, we will analyze the given absolute value equations and derive the possible values of \(|a-d|\).

### Step 1: Set up the equations based on absolute values
We have the following equations based on the problem statement:
1. \(|a - b| = 2\)
2. \(|b - c| = 3\)
3. \(|c - d| = 4\)

From these equations, we can express them as:
- \(a - b = 2\) or \(a - b = -2\)
- \(b - c = 3\) or \(b - c = -3\)
- \(c - d = 4\) or \(c - d = -4\)

### Step 2: Explore all combinations of signs
We will explore all combinations of the signs for the equations. This will give us different values for \(a - d\).

#### Case 1: All positive signs
- \(a - b = 2\)
- \(b - c = 3\)
- \(c - d = 4\)

Adding these gives:
\[ a - d = (a - b) + (b - c) + (c - d) = 2 + 3 + 4 = 9 \]

#### Case 2: All negative signs
- \(a - b = -2\)
- \(b - c = -3\)
- \(c - d = -4\)

Adding these gives:
\[ a - d = (-2) + (-3) + (-4) = -9 \]

#### Case 3: Two positive, one negative
- \(a - b = 2\)
- \(b - c = 3\)
- \(c - d = -4\)

Adding these gives:
\[ a - d = 2 + 3 - 4 = 1 \]

#### Case 4: One positive, two negative
- \(a - b = -2\)
- \(b - c = -3\)
- \(c - d = 4\)

Adding these gives:
\[ a - d = -2 - 3 + 4 = -1 \]

#### Case 5: One positive, one negative, one positive
- \(a - b = 2\)
- \(b - c = -3\)
- \(c - d = 4\)

Adding these gives:
\[ a - d = 2 - 3 + 4 = 3 \]

#### Case 6: One negative, one positive, one negative
- \(a - b = -2\)
- \(b - c = 3\)
- \(c - d = -4\)

Adding these gives:
\[ a - d = -2 + 3 - 4 = -3 \]

#### Case 7: Two negatives, one positive
- \(a - b = -2\)
- \(b - c = -3\)
- \(c - d = 4\)

Adding these gives:
\[ a - d = -2 - 3 + 4 = -1 \]

#### Case 8: Two positives, one negative
- \(a - b = 2\)
- \(b - c = 3\)
- \(c - d = -4\)

Adding these gives:
\[ a - d = 2 + 3 - 4 = 1 \]

### Step 3: Collect all possible values of \(|a - d|\)
From the cases, we have the following possible values for \(a - d\):
- \(9\)
- \(-9\)
- \(1\)
- \(-1\)
- \(3\)
- \(-3\)

Taking the absolute values, we have:
- \(|9| = 9\)
- \(|-9| = 9\)
- \(|1| = 1\)
- \(|-1| = 1\)
- \(|3| = 3\)
- \(|-3| = 3\)

Thus, the unique absolute values are:
- \(1, 3, 9\)

### Step 4: Sum all unique possible values of \(|a - d|\)
Now we sum the unique values:
\[ 1 + 3 + 9 = 13 \]

### Final Answer
The sum of all possible values of \(|a - d|\) is \(13\).