Question

What is the extraneous solution of 2|x−3|=4x ?(1 point)
x=

Answer 1

To solve the equation \( 2|x - 3| = 4x \), we first isolate the absolute value:

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

Next, we consider two cases for the absolute value expression:

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

\[
x - 3 = 2x
\]

Rearranging the equation, we get:

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

Since the solution \( x = -3 \) does not satisfy the condition \( x \geq 3 \), it is extraneous.

### Case 2: \( x - 3 < 0 \) (i.e., \( x < 3 \))
In this case, \( |x - 3| = -(x - 3) = 3 - x \). Substituting this into the equation gives:

\[
3 - x = 2x
\]

Rearranging the equation, we get:

\[
3 = 2x + x
\]
\[
3 = 3x
\]
\[
x = 1
\]

Since \( x = 1 \) satisfies the condition \( x < 3 \), it is a valid solution.

### Conclusion
The extraneous solution, which is the value that does not satisfy the condition for its case, is:

\[
\boxed{-3}
\]