https://file.aiquickdraw.com/mathbot/file/1727216027645ynhzbtjy.jpg

The equation shown in the image is:

\[
5|2n + 8| + 3 = 33
\]

This is an absolute value equation. To solve it, we proceed step by step:

### Step 1: Isolate the absolute value term
Subtract 3 from both sides of the equation:

\[
5|2n + 8| = 33 - 3
\]
\[
5|2n + 8| = 30
\]

### Step 2: Solve for the absolute value
Now, divide both sides by 5:

\[
|2n + 8| = \frac{30}{5}
\]
\[
|2n + 8| = 6
\]

### Step 3: Split into two cases
Since \( |2n + 8| = 6 \), we need to consider two cases:

#### Case 1: \( 2n + 8 = 6 \)
Solve for \( n \):
\[
2n = 6 - 8
\]
\[
2n = -2
\]
\[
n = \frac{-2}{2} = -1
\]

#### Case 2: \( 2n + 8 = -6 \)
Solve for \( n \):
\[
2n = -6 - 8
\]
\[
2n = -14
\]
\[
n = \frac{-14}{2} = -7
\]

### Final Answer:
The two possible values for \( n \) are \( n = -1 \) and \( n = -7 \).

---

Would you like further details on any part of this solution?

Here are some related questions:
1. What is the general approach for solving absolute value equations?
2. How do you handle absolute values in inequalities?
3. Can you apply a similar method to solve quadratic absolute value equations?
4. What are some special cases where absolute value equations have no solutions?
5. How do we verify the solutions in absolute value equations?

**Tip:** When solving absolute value equations, always split the equation into two cases, as the absolute value can represent both positive and negative values of the expression inside.

This guide provides a detailed step-by-step solution to the absolute value equation 5|2n + 8| + 3 = 33. Learn how to isolate the absolute value, solve for two cases, and find the values of n. Suitable for students in Grades 8-10.

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