Let's break down and clarify the equation you provided to solve it step by step. I believe there might be some ambiguity due to the formatting, so I'll interpret the equation as follows:

\[
1 + 3 \cdot \left| 2x + 1 \right| = 16
\]

Where:
- \(\left| 2x + 1 \right|\) represents the absolute value of \(2x + 1\).

### Steps to solve:

1. Start by isolating the absolute value expression:

\[
3 \cdot \left| 2x + 1 \right| = 16 - 1
\]
\[
3 \cdot \left| 2x + 1 \right| = 15
\]

2. Divide both sides by 3:

\[
\left| 2x + 1 \right| = \frac{15}{3} = 5
\]

3. The absolute value equation \(\left| 2x + 1 \right| = 5\) gives two possible cases:
   - Case 1: \(2x + 1 = 5\)
   - Case 2: \(2x + 1 = -5\)

4. Solve each case:

   - **Case 1**: 
   \[
   2x + 1 = 5
   \]
   Subtract 1 from both sides:
   \[
   2x = 4
   \]
   Divide by 2:
   \[
   x = 2
   \]

   - **Case 2**: 
   \[
   2x + 1 = -5
   \]
   Subtract 1 from both sides:
   \[
   2x = -6
   \]
   Divide by 2:
   \[
   x = -3
   \]

### Final Answer:
The solutions are \(x = 2\) and \(x = -3\).

---

Do you want further details on these steps or any clarification? 

Here are 5 related questions for practice:
1. Solve \(4 + 2 \cdot \left| x - 3 \right| = 10\).
2. Find \(x\) in the equation \(2\left| x + 1 \right| + 3 = 11\).
3. Solve \(\left| 5x - 2 \right| = 8\).
4. What are the solutions for \(x\) if \(\left| 3x + 4 \right| = 7\)?
5. Solve for \(x\) in \(1 + 2\left| 4x - 1 \right| = 13\).

**Tip**: When solving absolute value equations, always consider the positive and negative scenarios for the expression inside the absolute value.

Learn how to solve the absolute value equation 1 + 3|2x + 1| = 16 through a step-by-step breakdown. This problem highlights algebraic concepts such as absolute value equations and linear equations, making it suitable for students in Grades 8-10.

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