To solve the equation:

\[
|2 + 2n| + 1 = 3
\]

### Step 1: Isolate the absolute value
Subtract 1 from both sides:

\[
|2 + 2n| = 2
\]

### Step 2: Solve the absolute value equation
The absolute value equation \(|2 + 2n| = 2\) means:

\[
2 + 2n = 2 \quad \text{or} \quad 2 + 2n = -2
\]

#### Case 1: \(2 + 2n = 2\)
\[
2n = 0 \implies n = 0
\]

#### Case 2: \(2 + 2n = -2\)
\[
2n = -4 \implies n = -2
\]

### Solution
The solutions for \(n\) are:

\[
n = 0 \quad \text{and} \quad n = -2
\]

Would you like more details or have any questions?

Here are some related questions:
1. How do you solve more complex absolute value equations?
2. What happens if the absolute value equation has no solutions?
3. How do you solve absolute value inequalities?
4. What are the properties of absolute value functions?
5. Can absolute value equations have infinite solutions?

**Tip:** When solving absolute value equations, remember to check both the positive and negative cases.

Learn how to solve the absolute value equation |2 + 2n| + 1 = 3 with a step-by-step guide. This problem covers key algebraic concepts such as absolute value properties, isolating the absolute value, and solving both cases. Suitable for students in Grades 7-9.

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