The minimum value of `|3z-3|+|2z-4|`equal to

Options

A. 2
B. 1.5
C. 3
D. 1

Answer

To find the minimum value of the expression \( |3z - 3| + |2z - 4| \), we will follow these steps:

### Step 1: Rewrite the expression
We can factor out constants from the absolute values:
\[ |3z - 3| = 3|z - 1| \quad \text{and} \quad |2z - 4| = 2|z - 2| \]
Thus, we can rewrite the expression as:
\[ 3|z - 1| + 2|z - 2| \]

### Step 2: Analyze the absolute value terms
The expression \( |z - 1| \) represents the distance from \( z \) to 1, and \( |z - 2| \) represents the distance from \( z \) to 2. We want to minimize the total distance weighted by the coefficients 3 and 2.

### Step 3: Consider the points on the number line
We can visualize the points 1 and 2 on a number line. The minimum value of the expression occurs when \( z \) lies between these two points, as the distances will be minimized due to collinearity.

### Step 4: Set up the case for \( z \) between 1 and 2
If \( z \) is between 1 and 2, we can express the distances as:
\[ |z - 1| = z - 1 \quad \text{and} \quad |z - 2| = 2 - z \]
Substituting these into our expression gives:
\[ 3(z - 1) + 2(2 - z) = 3z - 3 + 4 - 2z = z + 1 \]

### Step 5: Find the minimum value
The expression \( z + 1 \) is minimized when \( z \) is at its minimum value in the interval [1, 2]. The minimum value occurs at \( z = 1 \):
\[ 1 + 1 = 2 \]

### Step 6: Check endpoints
We should also check the endpoints:
- At \( z = 1 \):
\[ 3|1 - 1| + 2|1 - 2| = 3(0) + 2(1) = 2 \]
- At \( z = 2 \):
\[ 3|2 - 1| + 2|2 - 2| = 3(1) + 2(0) = 3 \]

Thus, the minimum value occurs at \( z = 1 \) and is equal to 2.

### Conclusion
The minimum value of \( |3z - 3| + |2z - 4| \) is \( \boxed{2} \).

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