Question

What is the extraneous solution of 2|x-3| =4x?

Answer 1

To solve the equation \( 2|x-3| = 4x \), we start by isolating the absolute value expression:

\[
|x-3| = 2x
\]

Next, we need to consider the two cases for the absolute value:

### Case 1: \( x - 3 \geq 0 \) (i.e., \( x \geq 3 \))
In this case, \( |x-3| = x - 3 \). Thus, the equation becomes:

\[
x - 3 = 2x
\]

Rearranging gives:

\[
-x = 3 \quad \Rightarrow \quad x = -3
\]

However, \( x = -3 \) does not satisfy the condition \( x \geq 3 \). Therefore, there are no valid solutions from this case.

### Case 2: \( x - 3 < 0 \) (i.e., \( x < 3 \))
For this case, \( |x-3| = -(x - 3) = 3 - x \). Thus, our equation becomes:

\[
3 - x = 2x
\]

Rearranging gives:

\[
3 = 3x \quad \Rightarrow \quad x = 1
\]

Now we need to check if \( x = 1 \) satisfies the condition \( x < 3 \). Since \( 1 < 3 \), this solution is valid.

### Summary of Solutions
We found one valid solution \( x = 1 \).

### Identifying the Extraneous Solution
In the previous case analysis, we found \( x = -3 \) which is not a valid solution since it does not satisfy the condition for that case. Therefore, \( x = -3 \) is considered the extraneous solution.

In conclusion, the extraneous solution is:

\[
\boxed{-3}
\]