Which of the following is correct ?

Options

A. `7 + i gt 5 - i`
B. `8 + i gt 9 - i`
C. `13 + i lt 25 - 3i`
D. none of these

Answer

To determine which of the given inequalities is correct, we will analyze each option step by step.

### Step 1: Analyze the first inequality
**Inequality:** \( 7 + i > 5 - i \)

1. Rearrange the inequality by moving all real parts to one side and imaginary parts to the other:
\[ 7 + i + i > 5 \]
This simplifies to:
\[ 7 + 2i > 5 \]

2. Now, separate the real and imaginary parts:
- Real part: \( 7 > 5 \) (True)
- Imaginary part: \( 2i > 0 \) (This is not a valid comparison)

3. Since we cannot compare imaginary numbers directly, we conclude that this inequality is not valid.

### Step 2: Analyze the second inequality
**Inequality:** \( 8 + i > 9 - i \)

1. Rearranging gives:
\[ 8 + i + i > 9 \]
This simplifies to:
\[ 8 + 2i > 9 \]

2. Separate the real and imaginary parts:
- Real part: \( 8 > 9 \) (False)
- Imaginary part: \( 2i > 0 \) (Again, not a valid comparison)

3. Since the real part is false, this inequality is also not valid.

### Step 3: Analyze the third inequality
**Inequality:** \( 13 + i < 25 - 3i \)

1. Rearranging gives:
\[ 13 + i + 3i < 25 \]
This simplifies to:
\[ 13 + 4i < 25 \]

2. Separate the real and imaginary parts:
- Real part: \( 13 < 25 \) (True)
- Imaginary part: \( 4i < 0 \) (Again, not a valid comparison)

3. Since the real part is true, we need to check the imaginary part:
- We can express \( i \) as \( i < 0 \) which is false. However, since we are looking for the overall inequality, we can conclude that the real part holds.

### Conclusion
The only valid inequality among the three is:
\[ 13 + i < 25 - 3i \]
Thus, the correct answer is that the third inequality is true.

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