Question

What is the number of solutions to $|x - 2| = |x - 4|$? 
 

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

If $|x-a| = |x-b|$, the solution is always the midpoint $\frac{a+b}{2}$.

Answer 1

The Correct Option is A

- Step 1: Understanding the equation - The equation $|x - 2| = |x - 4|$ says the distance from $x$ to 2 is the same as the distance from $x$ to 4. 
- Step 2: Using the property of absolute values - The point that is equidistant from two numbers lies exactly at their midpoint. 
- Step 3: Finding the midpoint - Midpoint of 2 and 4 is: \[ \frac{2 + 4}{2} = 3 \] 
- Step 4: Conclusion from symmetry - The only value of $x$ satisfying the equation is $x = 3$. 
- Step 5: Verification - $|3 - 2| = 1$ and $|3 - 4| = 1$, so both sides are equal. 
- Step 6: Final answer - There is exactly 1 solution, matching option (1).