|x–1|+|x–3|+|x–5|+|x–7|=8

To solve the equation \( |x-1| + |x-3| + |x-5| + |x-7| = 8 \), we need to analyze it based on the critical points given by the absolute values, which are \( x = 1, 3, 5, 7 \).

We will consider different intervals determined by these points:

1. **Interval \( (-\infty, 1) \)**:
- Here, all absolute values are negative.
- The equation becomes:
\[
-(x-1) - (x-3) - (x-5) - (x-7) = 8
\]
- Simplifying gives:
\[
-4x + 16 = 8 \implies -4x = -8 \implies x = 2
\]
- Since \( 2 \) is not in this interval, there are no solutions here.

2. **Interval \( [1, 3) \)**:
- Here, \( |x-1| = x-1 \) and the others are still negative.
- The equation becomes:
\[
(x-1) - (x-3) - (x-5) - (x-7) = 8
\]
- Simplifying gives:
\[
-2x + 6 = 8 \implies -2x = 2 \implies x = -1
\]
- This is not in the interval, so no solutions here.

3. **Interval \( [3, 5) \)**:
- Here, \( |x-1| = x-1\), \( |x-3| = x-3\), and the others are negative.
- The equation becomes:
\[
(x-1) + (x-3) - (x-5) - (x-7) = 8
\]
- Simplifying gives:
\[
2x - 8 = 8 \implies 2x = 16 \implies x = 8
\]
- This is not in the interval, so no solutions here.

4. **Interval \( [5, 7) \)**:
- Here, \( |x-1| = x-1\), \( |x-3| = x-3\), \( |x-5| = x-5\), and the last is negative.
- The equation becomes:
\[
(x-1) + (x-3) + (x-5) - (x-7) = 8
\]
- Simplifying gives:
\[
3x - 14 = 8 \implies 3x = 22 \implies x = \frac{22}{3} \approx 7.33
\]
- This is not in the interval, so no solutions here.

5. **Interval \( [7, \infty) \)**:
- Here, all absolute values are positive.
- The equation becomes:
\[
(x-1) + (x-3) + (x-5) + (x-7) = 8
\]
- Simplifying gives:
\[
4x - 16 = 8 \implies 4x = 24 \implies x = 6
\]
- This does not satisfy \( x \geq 7\), so no solutions here.

After examining all intervals, we find that there are no \( x \) values that satisfy the original equation.

Thus, the solution to the equation \( |x-1| + |x-3| + |x-5| + |x-7| = 8 \) is:

**No solutions.**