The given inequality to solve is:

$|2x + 1| \leq 15$

To solve this absolute value inequality, we split it into two cases:

Step 1: Split the inequality

For absolute value inequalities, $|A| \leq B$ can be rewritten as: $-B \leq A \leq B$

Thus, we rewrite the original inequality: $-15 \leq 2x + 1 \leq 15$

Step 2: Solve for $x$

Left inequality:

$-15 \leq 2x + 1$ Subtract 1 from both sides: $-16 \leq 2x$ Divide by 2: $-8 \leq x$

Right inequality:

$2x + 1 \leq 15$ Subtract 1 from both sides: $2x \leq 14$ Divide by 2: $x \leq 7$

Step 3: Combine the results

The solution is: $-8 \leq x \leq 7$

Thus, the correct answer is the option: $\boxed{-8 \leq x \leq 7}$

Let me know if you'd like further clarification! Here are 5 related questions to expand your understanding:

1. How would the solution change if the inequality was $|2x + 1| > 15$?
2. Can you solve $|3x - 4| \leq 10$ using the same method?
3. How do we handle absolute value inequalities when the inequality is strict, such as $|A| < B$?
4. What are the key differences between solving absolute value equations and inequalities?
5. How can you graph the solution set $-8 \leq x \leq 7$ on a number line?

Tip: Always check for both cases in absolute value problems to ensure you're capturing the full solution set.