`{:("Column A",x = 3.636 cdot 10^(16), "Column B"),((x + 1)/(x - 1),,(x - 1)/(x+1)):}`

Options

A. If column A is larger
B. If column B is larger
C. If the columns are equal
D. If there is not enough information to decide

Answer

To solve the problem, we need to compare the values in Column A and Column B based on the given value of \( x = 3.636 \times 10^{6} \).

### Step 1: Define the expressions for Column A and Column B
- **Column A**: \( \frac{x + 1}{x - 1} \)
- **Column B**: \( \frac{x - 1}{x + 1} \)

### Step 2: Substitute the value of \( x \)
Substituting \( x = 3.636 \times 10^{6} \) into the expressions:
- Column A:
\[ \frac{3.636 \times 10^{6} + 1}{3.636 \times 10^{6} - 1} \]
- Column B:
\[ \frac{3.636 \times 10^{6} - 1}{3.636 \times 10^{6} + 1} \]

### Step 3: Analyze Column A
For Column A:
- The numerator \( x + 1 = 3.636 \times 10^{6} + 1 \) is greater than the denominator \( x - 1 = 3.636 \times 10^{6} - 1 \) because adding 1 to a positive number makes it larger, and subtracting 1 from a positive number makes it smaller.
- Therefore, \( \frac{x + 1}{x - 1} > 1 \).

### Step 4: Analyze Column B
For Column B:
- The numerator \( x - 1 = 3.636 \times 10^{6} - 1 \) is less than the denominator \( x + 1 = 3.636 \times 10^{6} + 1 \) for the same reasons as above.
- Therefore, \( \frac{x - 1}{x + 1} < 1 \).

### Step 5: Compare the two columns
From our analysis:
- Column A is greater than 1.
- Column B is less than 1.

Since Column A is greater than 1 and Column B is less than 1, it follows that:
\[ \text{Column A} > \text{Column B} \]

### Conclusion
Thus, we conclude that Column A is larger than Column B.

**Final Answer**: Column A is larger.

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