Find all solutions of the equation `|x - 1| -|x -8| = 3`

Answer

To solve the equation \( |x - 1| - |x - 8| = 3 \), we will analyze the absolute values by considering different cases based on the critical points where the expressions inside the absolute values change sign. The critical points here are \( x = 1 \) and \( x = 8 \).

### Step 1: Identify the cases based on critical points

1. **Case 1**: \( x < 1 \)
2. **Case 2**: \( 1 \leq x < 8 \)
3. **Case 3**: \( x \geq 8 \)

### Step 2: Solve each case

**Case 1**: \( x < 1 \)

In this case, both \( |x - 1| \) and \( |x - 8| \) will be negative:
\[ |x - 1| = -(x - 1) = -x + 1 \]
\[ |x - 8| = -(x - 8) = -x + 8 \]

Substituting these into the equation:
\[ -x + 1 - (-x + 8) = 3 \]
This simplifies to:
\[ -x + 1 + x - 8 = 3 \]
\[ 1 - 8 = 3 \]
\[ -7 = 3 \]
This is a contradiction, so there are no solutions in this case.

**Case 2**: \( 1 \leq x < 8 \)

Here, \( |x - 1| \) is positive and \( |x - 8| \) is negative:
\[ |x - 1| = x - 1 \]
\[ |x - 8| = -(x - 8) = -x + 8 \]

Substituting these into the equation:
\[ (x - 1) - (-x + 8) = 3 \]
This simplifies to:
\[ x - 1 + x - 8 = 3 \]
\[ 2x - 9 = 3 \]
Adding 9 to both sides:
\[ 2x = 12 \]
Dividing by 2:
\[ x = 6 \]
Since \( 6 \) is in the interval \( [1, 8) \), it is a valid solution.

**Case 3**: \( x \geq 8 \)

In this case, both \( |x - 1| \) and \( |x - 8| \) are positive:
\[ |x - 1| = x - 1 \]
\[ |x - 8| = x - 8 \]

Substituting these into the equation:
\[ (x - 1) - (x - 8) = 3 \]
This simplifies to:
\[ x - 1 - x + 8 = 3 \]
\[ 7 = 3 \]
This is also a contradiction, so there are no solutions in this case.

### Conclusion

The only solution to the equation \( |x - 1| - |x - 8| = 3 \) is:
\[ \boxed{6} \]