Solve the equation. \[ |x-2|=|x+2| \]

To solve the equation \( |x-2|=|x+2| \), we need to consider the possible cases for the absolute values. **Case 1:** \( x - 2 = x + 2 \) Subtracting \( x \) from both sides, we get: \[ -2 = 2 \] This is not possible, so there are no solutions in this case. **Case 2:** \( x - 2 = -(x + 2) \) This can be rewritten as: \[ x - 2 = -x - 2 \] Adding \( x \) to both sides gives: \[ 2x - 2 = -2 \] Now adding 2 to both sides: \[ 2x = 0 \] Dividing by 2: \[ x = 0 \] **Case 3:** \( -(x - 2) = x + 2 \) This can be rewritten as: \[ - x + 2 = x + 2 \] Adding \( x \) to both sides results in: \[ 2 = 2x + 2 \] Subtracting 2 from both sides gives: \[ 0 = 2x \] Dividing by 2: \[ x = 0 \] **Conclusion:** The only solution to the equation \( |x-2|=|x+2| \) is: \[ \boxed{0} \]