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{:("Column A", x > 1,"Column B"),(1//x,"...

`{:("Column A", x > 1,"Column B"),(1//x," ",1/(x - 1)):}`

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

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to compare the values in Column A and Column B given that \( x > 1 \). ### Step-by-step Solution: 1. **Identify the expressions in both columns**: - Column A: \( \frac{1}{x} \) - Column B: \( \frac{1}{x - 1} \) 2. **Understand the condition**: - We know that \( x > 1 \). This means that \( x - 1 > 0 \) as well. 3. **Analyze the denominators**: - The denominator of Column A is \( x \). - The denominator of Column B is \( x - 1 \). - Since \( x > 1 \), we can conclude that \( x - 1 < x \). 4. **Compare the fractions**: - Since \( x - 1 < x \), it follows that \( \frac{1}{x - 1} > \frac{1}{x} \). This is because for fractions, a smaller denominator results in a larger value. 5. **Conclusion**: - Therefore, \( \frac{1}{x - 1} > \frac{1}{x} \) implies that Column B is greater than Column A for any \( x > 1 \). ### Final Answer: - Column B is larger than Column A.
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