What is the sum of all the possible values of x if |x - 1| + Ix - 7| = 10?

Options

A. 0
B. 3
C. 6
D. 8

Answer

To solve the equation \( |x - 1| + |x - 7| = 10 \), we need to consider the different cases that arise from the absolute values.

### Step 1: Identify the critical points
The critical points for the absolute values are \( x = 1 \) and \( x = 7 \). These points divide the number line into three intervals:
1. \( x < 1 \)
2. \( 1 \leq x < 7 \)
3. \( x \geq 7 \)

### Step 2: Solve for each interval

**Case 1: \( x < 1 \)**
In this interval, both expressions inside the absolute values are negative:
\[ |x - 1| = -(x - 1) = -x + 1 \]
\[ |x - 7| = -(x - 7) = -x + 7 \]
Thus, the equation becomes:
\[ (-x + 1) + (-x + 7) = 10 \]
Simplifying this gives:
\[ -2x + 8 = 10 \]
\[ -2x = 10 - 8 \]
\[ -2x = 2 \implies x = -1 \]
Since \(-1 < 1\), this solution is valid.

**Case 2: \( 1 \leq x < 7 \)**
In this interval, \( |x - 1| \) is positive and \( |x - 7| \) is negative:
\[ |x - 1| = x - 1 \]
\[ |x - 7| = -(x - 7) = -x + 7 \]
Thus, the equation becomes:
\[ (x - 1) + (-x + 7) = 10 \]
Simplifying this gives:
\[ -1 + 7 = 10 \]
\[ 6 = 10 \]
This is not true, so there are no solutions in this interval.

**Case 3: \( x \geq 7 \)**
In this interval, both expressions inside the absolute values are positive:
\[ |x - 1| = x - 1 \]
\[ |x - 7| = x - 7 \]
Thus, the equation becomes:
\[ (x - 1) + (x - 7) = 10 \]
Simplifying this gives:
\[ 2x - 8 = 10 \]
\[ 2x = 10 + 8 \]
\[ 2x = 18 \implies x = 9 \]
Since \(9 \geq 7\), this solution is valid.

### Step 3: Collect all valid solutions
The valid solutions we found are:
1. \( x = -1 \)
2. \( x = 9 \)

### Step 4: Calculate the sum of all possible values of \( x \)
Now, we find the sum of all possible values:
\[ -1 + 9 = 8 \]

### Final Answer
The sum of all the possible values of \( x \) is \( \boxed{8} \).