To solve the inequality \( |x + 5| \leq 8 \), we will break it down based on the definition of absolute value.
The inequality \( |A| \leq B \) means that \( -B \leq A \leq B \). In this case:
\[ -8 \leq x + 5 \leq 8. \]
We will solve the two inequalities separately.
- Solve the left inequality:
\[ -8 \leq x + 5 \]
Subtract 5 from both sides:
\[ -8 - 5 \leq x \implies -13 \leq x \implies x \geq -13. \]
- Solve the right inequality:
\[ x + 5 \leq 8. \]
Subtract 5 from both sides:
\[ x \leq 8 - 5 \implies x \leq 3. \]
Combining the results from both inequalities, we find:
\[ -13 \leq x \leq 3. \]
Thus, the solution to the inequality \( |x + 5| \leq 8 \) is:
\[ \boxed{[-13, 3]}. \]