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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.
1. Solve the left inequality:
\[
-8 \leq x + 5
\]
Subtract 5 from both sides:
\[
-8 - 5 \leq x \implies -13 \leq x \implies x \geq -13.
\]
2. 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:
\[
\boxed{[-13, 3]}.
\]