|x+5|+|x+1|=7

To solve the equation \(|x+5| + |x+1| = 7\), we need to consider the critical points where the expressions inside the absolute values change sign. The critical points for this equation are \(x = -5\) and \(x = -1\). We will consider three cases based on these points:

### Case 1: \(x < -5\)

In this case, both \(x + 5\) and \(x + 1\) are negative, so we have:
\[
|x+5| = -(x+5) = -x - 5,
\]
\[
|x+1| = -(x+1) = -x - 1.
\]
Substituting these into the equation gives:
\[
(-x - 5) + (-x - 1) = 7,
\]
which simplifies to:
\[
-2x - 6 = 7.
\]
Solving for \(x\):
\[
-2x = 7 + 6,
\]
\[
-2x = 13,
\]
\[
x = -\frac{13}{2} = -6.5.
\]
Since \(-6.5 < -5\), this solution is valid in this interval.

### Case 2: \(-5 \leq x < -1\)

In this case, \(x + 5\) is non-negative and \(x + 1\) is negative. Therefore:
\[
|x+5| = x + 5,
\]
\[
|x+1| = -(x+1) = -x - 1.
\]
Substituting these into the equation gives:
\[
(x + 5) + (-x - 1) = 7,
\]
which simplifies to:
\[
5 - 1 = 7,
\]
\[
4 = 7.
\]
This is not true, so there are no solutions in this interval.

### Case 3: \(x \geq -1\)

In this case, both \(x + 5\) and \(x + 1\) are non-negative. Therefore:
\[
|x+5| = x + 5,
\]
\[
|x+1| = x + 1.
\]
Substituting these into the equation gives:
\[
(x + 5) + (x + 1) = 7,
\]
which simplifies to:
\[
2x + 6 = 7.
\]
Solving for \(x\):
\[
2x = 7 - 6,
\]
\[
2x = 1,
\]
\[
x = \frac{1}{2}.
\]
Since \(\frac{1}{2} \geq -1\), this solution is valid in this interval.

### Summary of solutions

The solutions to the equation \(|x+5| + |x+1| = 7\) are:
\[
x = -6.5 \quad \text{and} \quad x = \frac{1}{2}.
\]

Therefore, the final answer is:
\[
x = -\frac{13}{2}, \quad \text{and} \quad x = \frac{1}{2}.
\]