Find the extraneous solution of the equation |x - 8| = 3x
Find x

To solve the equation \( |x - 8| = 3x \), we need to consider two cases, since the absolute value equation can be split based on the definition of absolute value:

### Case 1: \( x - 8 = 3x \)
Here, the expression inside the absolute value is non-negative.

1. Subtract \( x \) from both sides:
   \[
   -8 = 2x
   \]

2. Solve for \( x \):
   \[
   x = -4
   \]

### Case 2: \( -(x - 8) = 3x \)
This handles the case where the expression inside the absolute value is negative.

1. Simplify the equation:
   \[
   -x + 8 = 3x
   \]

2. Add \( x \) to both sides:
   \[
   8 = 4x
   \]

3. Solve for \( x \):
   \[
   x = 2
   \]

### Now, check both solutions in the original equation:
- For \( x = -4 \):
  \[
  |-4 - 8| = | -12 | = 12
  \]
  On the other hand:
  \[
  3(-4) = -12
  \]
  So, \( |x - 8| = 3x \) holds, making \( x = -4 \) a valid solution.

- For \( x = 2 \):
  \[
  |2 - 8| = |-6| = 6
  \]
  And:
  \[
  3(2) = 6
  \]
  So, \( |x - 8| = 3x \) also holds, making \( x = 2 \) a valid solution as well.

### Extraneous Solution:
There is **no extraneous solution** in this case, as both solutions \( x = -4 \) and \( x = 2 \) are valid.

Would you like more details or clarification?

Here are 5 related questions you could explore:
1. What is the general approach for solving absolute value equations?
2. How do you determine whether a solution is extraneous?
3. Can an equation with absolute values have multiple extraneous solutions?
4. How does graphing help in understanding absolute value equations?
5. What happens when both sides of an equation contain absolute values?

**Tip:** Always check your solutions in the original equation when dealing with absolute values to rule out extraneous solutions.

Find the extraneous solution of the equation |x - 8| = 3x. Find x.

Learn how to find the extraneous solution of the absolute value equation |x - 8| = 3x. This problem explores key algebraic concepts and properties of absolute values, suitable for students in Grades 8-10.

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