The inequality `|z-4| < |z-2|` represents

Options

A. Re(z) `lt 0`
B. Re(z) `gt 0`
C. Re(z) `gt 2`
D. Re(z) `gt 3`

Answer

To solve the inequality \( |z - 4| < |z - 2| \), we will follow these steps:

### Step 1: Rewrite the inequality
We start with the given inequality:
\[ |z - 4| < |z - 2| \]

### Step 2: Interpret the inequality geometrically
The expression \( |z - 4| \) represents the distance from the complex number \( z \) to the point \( 4 + 0i \) (which is the point \( (4, 0) \) on the Argand plane), and \( |z - 2| \) represents the distance from \( z \) to the point \( 2 + 0i \) (the point \( (2, 0) \)).

### Step 3: Identify the points on the Argand plane
We plot the points \( (4, 0) \) and \( (2, 0) \) on the Argand plane. The inequality states that the distance from \( z \) to \( (4, 0) \) must be less than the distance from \( z \) to \( (2, 0) \).

### Step 4: Find the midpoint
The midpoint between the points \( (4, 0) \) and \( (2, 0) \) is:
\[ \left( \frac{4 + 2}{2}, 0 \right) = (3, 0) \]
This means that the line \( x = 3 \) (where the real part of \( z \) is 3) bisects the segment between these two points.

### Step 5: Determine the region
Since we want the distance to \( (4, 0) \) to be less than the distance to \( (2, 0) \), the region satisfying this inequality will be to the left of the line \( x = 3 \). Thus, we can express this condition as:
\[ \text{Re}(z) < 3 \]

### Step 6: Conclusion
The inequality \( |z - 4| < |z - 2| \) represents the region in the complex plane where the real part of \( z \) is less than 3.

### Final Result
The solution can be summarized as:
\[ \text{The region represented by } |z - 4| < |z - 2| \text{ is } \text{Re}(z) < 3. \]

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