|x+5|=3; |x−4|=1; |x+8|=7

To solve the equations involving absolute values, we can break each equation into cases based on the definition of absolute value.

1. \( |x+5| = 3 \)

This breaks down into two cases:

Case 1:
\( x + 5 = 3 \)
\( x = 3 - 5 \)
\( x = -2 \)

Case 2:
\( x + 5 = -3 \)
\( x = -3 - 5 \)
\( x = -8 \)

So, the solutions for \( |x + 5| = 3 \) are \( x = -2 \) and \( x = -8 \).

2. \( |x−4| = 1 \)

This also breaks down into two cases:

Case 1:
\( x - 4 = 1 \)
\( x = 1 + 4 \)
\( x = 5 \)

Case 2:
\( x - 4 = -1 \)
\( x = -1 + 4 \)
\( x = 3 \)

So, the solutions for \( |x - 4| = 1 \) are \( x = 5 \) and \( x = 3 \).

3. \( |x+8| = 7 \)

This again breaks down into two cases:

Case 1:
\( x + 8 = 7 \)
\( x = 7 - 8 \)
\( x = -1 \)

Case 2:
\( x + 8 = -7 \)
\( x = -7 - 8 \)
\( x = -15 \)

So, the solutions for \( |x + 8| = 7 \) are \( x = -1 \) and \( x = -15 \).

Summary of Solutions

1. For \( |x + 5| = 3 \): \( x = -2, -8 \)
2. For \( |x - 4| = 1 \): \( x = 5, 3 \)
3. For \( |x + 8| = 7 \): \( x = -1, -15 \)