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{:("Column A",0 < x < 1, "Column B"),(x^...

`{:("Column A",0 < x < 1, "Column B"),(x^2,,"x"):}`

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 the condition \(0 < x < 1\). ### Step-by-Step Solution: 1. **Identify the expressions in each column**: - Column A: \(x^2\) - Column B: \(x\) 2. **Understand the range of \(x\)**: - We know that \(x\) is between 0 and 1, which means \(0 < x < 1\). 3. **Analyze the relationship between \(x\) and \(x^2\)**: - Since \(x\) is a positive number less than 1, squaring \(x\) will yield a smaller number. This is because multiplying a fraction (which is less than 1) by itself results in a smaller fraction. - Mathematically, for \(0 < x < 1\), we can say: \[ x^2 < x \] 4. **Test with specific values**: - Let's take a few values of \(x\) within the range (0, 1): - If \(x = 0.1\): - Column A: \(0.1^2 = 0.01\) - Column B: \(0.1\) - Here, \(0.1 > 0.01\) (Column B > Column A). - If \(x = 0.5\): - Column A: \(0.5^2 = 0.25\) - Column B: \(0.5\) - Here, \(0.5 > 0.25\) (Column B > Column A). - If \(x = 0.9\): - Column A: \(0.9^2 = 0.81\) - Column B: \(0.9\) - Here, \(0.9 > 0.81\) (Column B > Column A). 5. **Conclusion**: - In all cases tested, we find that \(x > x^2\) for any value of \(x\) in the interval \(0 < x < 1\). - Therefore, we conclude that Column B is always greater than Column A. ### Final Answer: - Column B is larger than Column A.
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