x is an integer, and r is the least possible value of sqrt((4x - 7)^2) + 12. Quantity A: r, Quantity B: 3.5

Rationale

Quantity A is greater.

To determine the least possible value of \( r = \sqrt{(4x - 7)^2} + 12 \), we observe that \( \sqrt{(4x - 7)^2} \) simplifies to \( |4x - 7| \). The minimum value of \( |4x - 7| \) occurs when \( 4x - 7 = 0 \), giving us \( r = 12 \). Since \( 12 \) is greater than \( 3.5 \), Quantity A is indeed greater.

A (Quantity A is greater.) CORRECT

This choice accurately reflects that the minimum value of \( r \) is \( 12 \), which is clearly greater than \( 3.5 \). Therefore, this option is correct and aligns with our calculation.

B (Quantity B is greater.) YOUR ANSWER

This choice suggests that \( 3.5 \) is greater than \( r \). However, since we established that the least value of \( r \) is \( 12 \), this statement is incorrect because \( 12 \) is greater than \( 3.5 \).

C (The two quantities are equal.) YOUR ANSWER

This option implies \( r \) equals \( 3.5 \). Given that the minimum value of \( r \) is \( 12 \), this cannot be true. Therefore, this choice is also incorrect.

D (The relationship cannot be determined from the information given.) YOUR ANSWER

This choice suggests that we lack sufficient information to compare the two quantities. However, we have explicitly calculated the minimum value of \( r \) as \( 12 \), allowing for a clear comparison with \( 3.5 \). Thus, this option is inaccurate.

Conclusion

The analysis reveals that the least possible value of \( r \) is \( 12 \), which is greater than \( 3.5 \). This confirms that Quantity A is greater than Quantity B. The other options do not represent valid comparisons based on the information provided and calculations performed.

x is an integer, and r is the least possible value of sqrt((4x - 7)^2) + 12. Quantity A: r, Quantity B: 3.5

Quantity A is greater.

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