https://file.aiquickdraw.com/mathbot/file/1726897132026jj2npokj.jpg

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.

Learn how to solve the absolute value inequality |2x + 1| ≤ 15 step-by-step. This guide covers key algebraic concepts, including rewriting inequalities and solving for x. Ideal for students in Grades 8-10.

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