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Determine which of the two expressions b...

Determine which of the two expressions below is larger, whether they are equal, or whether there is not enough information to decide. [The answer is A if Column A is larger, B is Column B is larger ,C if the Columns are equal and D if there is not enough information to decide]
`xne0`
Column A ` -->` `x`
Column B `-->` `x^2`

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 determine which of the two expressions is larger, we will analyze the values of \( x \) in both columns. We have: - Column A: \( x \) - Column B: \( x^2 \) Given that \( x \neq 0 \), we can consider different cases for \( x \): positive values, negative values, and fractions. ### Step 1: Analyze for positive values of \( x \) Let’s first consider when \( x \) is positive. - If \( x = 1 \): - Column A: \( x = 1 \) - Column B: \( x^2 = 1^2 = 1 \) - Conclusion: Column A = Column B - If \( x = 2 \): - Column A: \( x = 2 \) - Column B: \( x^2 = 2^2 = 4 \) - Conclusion: Column B > Column A ### Step 2: Analyze for negative values of \( x \) Next, let’s consider when \( x \) is negative. - If \( x = -1 \): - Column A: \( x = -1 \) - Column B: \( x^2 = (-1)^2 = 1 \) - Conclusion: Column B > Column A ### Step 3: Analyze for fractional positive values of \( x \) Now, let’s consider when \( x \) is a positive fraction. - If \( x = \frac{1}{2} \): - Column A: \( x = \frac{1}{2} \) - Column B: \( x^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \) - Conclusion: Column A > Column B ### Summary of Findings From our analysis, we have: - For \( x = 1 \): Column A = Column B - For \( x = 2 \): Column B > Column A - For \( x = -1 \): Column B > Column A - For \( x = \frac{1}{2} \): Column A > Column B ### Final Conclusion Since the relationship between Column A and Column B changes depending on the value of \( x \), we conclude that there is not enough information to decide which column is consistently larger. **Final Answer: D (There is not enough information to decide)** ---
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