Solve for x.
`abs(x-3)+abs(4-x)=1`

Answer

To solve the equation \( |x - 3| + |4 - x| = 1 \), we will consider different cases based on the values of \( x \).

### Step 1: Identify the critical points
The expression involves absolute values, so we need to identify the critical points where the expressions inside the absolute values change signs. The critical points here are \( x = 3 \) and \( x = 4 \).

### Step 2: Break down into cases
We will consider three cases based on the critical points:

1. **Case 1:** \( x < 3 \)
2. **Case 2:** \( 3 \leq x < 4 \)
3. **Case 3:** \( x \geq 4 \)

### Case 1: \( x < 3 \)
In this case, both expressions inside the absolute values are negative:
- \( |x - 3| = -(x - 3) = 3 - x \)
- \( |4 - x| = -(4 - x) = x - 4 \)

Substituting these into the equation:
\[ (3 - x) + (x - 4) = 1 \]
Simplifying:
\[ 3 - x + x - 4 = 1 \\ -1 = 1 \]
This is a contradiction, so there are no solutions in this case.

### Case 2: \( 3 \leq x < 4 \)
In this case:
- \( |x - 3| = x - 3 \)
- \( |4 - x| = 4 - x \)

Substituting these into the equation:
\[ (x - 3) + (4 - x) = 1 \]
Simplifying:
\[ x - 3 + 4 - x = 1 \\ 1 = 1 \]
This is always true, which means any \( x \) in the interval \( [3, 4) \) is a solution.

### Case 3: \( x \geq 4 \)
In this case:
- \( |x - 3| = x - 3 \)
- \( |4 - x| = -(4 - x) = x - 4 \)

Substituting these into the equation:
\[ (x - 3) + (x - 4) = 1 \]
Simplifying:
\[ 2x - 7 = 1 \\ 2x = 8 \\ x = 4 \]
This solution is valid since \( x = 4 \) is included in this case.

### Final Solution
Combining the results from all cases, we find:
- From Case 1: No solutions.
- From Case 2: All \( x \) in the interval \( [3, 4) \) are solutions.
- From Case 3: The solution \( x = 4 \).

Thus, the complete solution is:
\[ x \in [3, 4] \]