To solve the equation \( |x - 8| = 3x \), we need to consider the definition of the absolute value and analyze both possible cases:

1. \( x - 8 = 3x \) (the expression inside the absolute value is non-negative)
2. \( x - 8 = -3x \) (the expression inside the absolute value is negative)

Let's solve each case separately.

### Case 1: \( x - 8 = 3x \)

\[
x - 8 = 3x
\]

Subtract \( x \) from both sides:

\[
-8 = 2x
\]

Divide by 2:

\[
x = -4
\]

Now, we need to check if this solution satisfies the condition for which this case was considered. This case assumes \( x - 8 \geq 0 \), which implies \( x \geq 8 \). Since \( x = -4 \) does not satisfy \( x \geq 8 \), this is **not a valid solution** for this case.

### Case 2: \( x - 8 = -3x \)

\[
x - 8 = -3x
\]

Add \( 3x \) to both sides:

\[
x + 3x = 8
\]

Combine like terms:

\[
4x = 8
\]

Divide by 4:

\[
x = 2
\]

This case assumes \( x - 8 < 0 \), which implies \( x < 8 \). Since \( x = 2 \) satisfies \( x < 8 \), this **is a valid solution**.

### Conclusion

The only valid solution for the equation \( |x - 8| = 3x \) is:

\[
x = 2
\]

Would you like more details on these steps, or do you have any other questions?

Here are some related questions you might find interesting:

1. How do you solve absolute value equations where the variable appears on both sides?
2. What are some examples of absolute value equations with no real solutions?
3. How can you check if an absolute value equation has valid solutions after solving?
4. What is the geometric interpretation of an absolute value equation like \( |x - 8| = 3x \)?
5. How do you solve absolute value inequalities, and how do they differ from equations?

**Tip:** Always check each potential solution to ensure it meets the conditions set by the original absolute value equation.

Learn how to solve the absolute value equation |x - 8| = 3x with a detailed step-by-step solution. This problem covers concepts like absolute value equations and linear equations. Suitable for students in Grades 9-12.

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