Question 12

Quantity A is greater.

The least possible value of \( \sqrt{(4x - 7)^2} + 12 \) occurs when \( \sqrt{(4x - 7)^2} \) is minimized. The term \( \sqrt{(4x - 7)^2} \) reaches its minimum value of 0 when \( 4x - 7 = 0 \), which means \( x = 1.75 \). However, since \( x \) must be an integer, the closest integers are 1 and 2, leading to a minimum value of \( 12 \) when \( x = 2 \), which is greater than \( 3.5 \).

A) Quantity A is greater.

This option is correct because the minimum value of \( r \) calculated as \( 12 \) when \( x = 2 \) is indeed greater than \( 3.5 \).

B) Quantity B is greater.

This option is incorrect because \( 3.5 \) is significantly less than the minimum value of \( r \) calculated to be \( 12 \). Therefore, Quantity B does not exceed Quantity A.

C) The two quantities are equal.

This option is incorrect since the minimum value of Quantity A is \( 12 \), which is not equal to \( 3.5 \). Thus, there is a definitive difference between the two quantities.

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

This option is incorrect because the calculations show a clear relationship between the two quantities. The least possible value of Quantity A can be determined, confirming that it is greater than Quantity B.

Conclusion

The minimum value of \( r \) is \( 12 \), which is clearly greater than \( 3.5 \). All other options fail to accurately represent the relationship between the two quantities, as they either misinterpret or overlook the calculated minimum value. Therefore, Quantity A is definitively greater than Quantity B.