The inequality `a + ib gt c + id` is true when

Options

A. `a gt c, b gt d gt 0`
B. `a gt c, b = d = 0`
C. `a gt c, b = d gt 0`
D. None of these

Answer

To solve the inequality \( a + ib > c + id \) for complex numbers, we need to analyze the components of the complex numbers involved.

### Step-by-Step Solution:

1. **Understanding the Inequality**:
The inequality \( a + ib > c + id \) implies that we are comparing two complex numbers. Here, \( a \) and \( c \) are the real parts, while \( b \) and \( d \) are the imaginary parts of the complex numbers.

2. **Separating Real and Imaginary Parts**:
We can separate the inequality into its real and imaginary components:
\[ a > c \quad \text{and} \quad b > d \]
However, the comparison of imaginary parts does not affect the inequality of complex numbers in the same way it does for real numbers.

3. **Conditions for the Inequality**:
The inequality \( a + ib > c + id \) holds true primarily based on the comparison of the real parts. The imaginary parts do not contribute to the inequality in the same manner as real numbers. Thus, we can conclude:
- The inequality \( a + ib > c + id \) is true if and only if:
\[ a > c \quad \text{and} \quad b = d \]

4. **Conclusion**:
Therefore, the inequality \( a + ib > c + id \) is true when:
- \( a > c \) (the real part of the first complex number is greater than the real part of the second)
- \( b = d \) (the imaginary parts are equal)

### Final Answer:
The inequality \( a + ib > c + id \) is true when \( a > c \) and \( b = d \).