https://file.aiquickdraw.com/mathbot/file/1726758265093vttdfyon.jpg

The equation shown in the image is:

\[
|8 - 2n| - 5n = n + 2
\]

To solve this equation involving absolute value, we need to break it into two possible cases based on the expression inside the absolute value:

### Case 1: \( 8 - 2n \geq 0 \) or \( n \leq 4 \)
In this case, the absolute value of \( 8 - 2n \) is simply \( 8 - 2n \). So the equation becomes:
\[
8 - 2n - 5n = n + 2
\]
Simplify:
\[
8 - 7n = n + 2
\]
Move all terms involving \( n \) to one side:
\[
8 - 2 = n + 7n
\]
\[
6 = 8n
\]
Solve for \( n \):
\[
n = \frac{6}{8} = \frac{3}{4}
\]
This solution is valid because \( n = \frac{3}{4} \leq 4 \).

### Case 2: \( 8 - 2n < 0 \) or \( n > 4 \)
In this case, the absolute value of \( 8 - 2n \) is \( -(8 - 2n) = -8 + 2n \). So the equation becomes:
\[
-8 + 2n - 5n = n + 2
\]
Simplify:
\[
-8 - 3n = n + 2
\]
Move all terms involving \( n \) to one side:
\[
-8 - 2 = n + 3n
\]
\[
-10 = 4n
\]
Solve for \( n \):
\[
n = \frac{-10}{4} = -\frac{5}{2}
\]
However, this solution is invalid because \( n = -\frac{5}{2} \) does not satisfy \( n > 4 \). Hence, we discard this solution.

### Final Answer:
The only valid solution is \( n = \frac{3}{4} \).

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How do we graph absolute value functions?
2. Can you find the solution if the equation was \( |8 - 2n| + 5n = n + 2 \)?
3. What happens if the inequality for the second case is reversed?
4. How does the solution change if the constant term on the right side is different?
5. How can we check if the solution is correct by substituting into the original equation?

**Tip:** When solving absolute value equations, always check both cases to ensure that the solution satisfies the constraints set by the absolute value function.

Discover how to solve the absolute value equation |8 - 2n| - 5n = n + 2 with a comprehensive step-by-step guide. This problem involves algebraic techniques and properties of absolute values, ideal for students in Grades 8-10.

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