Solve and check :`|x-3|=2x`.

Answer

To solve the equation \( |x - 3| = 2x \), we need to consider the two cases that arise from the absolute value.

### Step 1: Set up the two cases for the absolute value
The absolute value equation \( |x - 3| = 2x \) can be split into two cases:
1. Case 1: \( x - 3 = 2x \)
2. Case 2: \( x - 3 = -2x \)

### Step 2: Solve Case 1
For Case 1, we have:
\[ x - 3 = 2x \]
Subtract \( x \) from both sides:
\[ -3 = 2x - x \]
This simplifies to:
\[ -3 = x \]

### Step 3: Solve Case 2
For Case 2, we have:
\[ x - 3 = -2x \]
Add \( 2x \) to both sides:
\[ x + 2x - 3 = 0 \]
This simplifies to:
\[ 3x - 3 = 0 \]
Now, add 3 to both sides:
\[ 3x = 3 \]
Divide by 3:
\[ x = 1 \]

### Step 4: Summary of solutions
The two potential solutions we found are:
1. \( x = -3 \)
2. \( x = 1 \)

### Step 5: Check the solutions
Now we need to check both solutions in the original equation \( |x - 3| = 2x \).

**Checking \( x = -3 \):**
\[ |-3 - 3| = 2(-3) \]
This simplifies to:
\[ |-6| = -6 \]
Which gives:
\[ 6 \neq -6 \quad \text{(not valid)} \]

**Checking \( x = 1 \):**
\[ |1 - 3| = 2(1) \]
This simplifies to:
\[ |-2| = 2 \]
Which gives:
\[ 2 = 2 \quad \text{(valid)} \]

### Final Solution
The only valid solution is:
\[ \boxed{1} \]

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