Question

The minimum value of \( |z - 1| + |z - 5| \) is:

• A. \( 3 \)
• B. \( 5 \)
• C. \( 4 \)
• D. \( 2 \)

Hint:

To minimize expressions of the form \( |z - a| + |z - b| \), choose \( z \) as the median of \( a \) and \( b \). This concept is especially useful in problems involving absolute values and optimization.

Answer 1

The Correct Option is C

Step 1: We are given the expression \( |z - 1| + |z - 5| \). This represents the sum of distances from a point \( z \) to the two fixed points 1 and 5 on the number line.
Step 2: To find the minimum value of such an expression, we use the following key fact: For two real numbers \( a \) and \( b \), the expression \( |x - a| + |x - b| \) is minimized when \( z \) lies between \( a \) and \( b \), and more specifically, at the median of \( a \) and \( b \).
Step 3: In this case, the two points are \( a = 1 \) and \( b = 5 \). The median (middle value) of 1 and 5 is: \[ x = \frac{1 + 5}{2} = 3. \] Step 4: Substitute \( z = 3 \) into the expression to find the minimum value: \[ |3 - 1| + |3 - 5| = 2 + 2 = 4. \] Therefore, the minimum value is \( \boxed{4} \).