11. Which of the following expressions is equivalent to the expression 3(a - 4) - 2(b + 8) for all values of a and b?
Answer: D
3(a - 4) - 2(b + 8) simplifies to 3a - 2b - 28
To determine the equivalent expression, we first distribute the terms in the expression 3(a - 4) - 2(b + 8). This results in 3a - 12 - 2b - 16, which simplifies to 3a - 2b - 28.
A) 3a - 2b + 4
This option is incorrect because it suggests that the expression results in a positive constant term. The original expression simplifies to a negative constant, not a positive one.
B) 3a - 2b - 4
This option is also incorrect. Although it maintains the correct variable terms, it inaccurately represents the constant term, which should be -28 rather than -4.
C) 3a - 2b - 20
This option is incorrect as well. While it correctly includes the variable terms, the constant term is miscalculated; the expression should yield -28, not -20.
D) 3a - 2b - 28
This is the correct option. It accurately reflects the result of simplifying the original expression, with the constant term correctly calculated as -28.
Conclusion
The correct answer is definitively D because it accurately represents the simplified form of the expression 3(a - 4) - 2(b + 8). All other options fail as they either miscalculate the constant or do not correctly represent the relationship between the variables a and b. Thus, D is the only expression that remains valid for all values of a and b.