Final answer:
The statement that must be true is D. c + a < c - a, which logically follows from the fact that a is a negative number. Adding a negative number will always result in a smaller value than subtracting the same negative number.
Step-by-step explanation:
The question asked which statement must be true given that a, b, c, and d are integers and a < 0. To find the correct statement, we analyze them one by one considering a is negative:
- A. b - a > c - a: This statement compares two expressions which subtract the same negative number from two undetermined integers. Since we don't have information about the relationship between b and c, we cannot determine the truth of this statement.
- B. b - a < c - a: This has the same issue as option A.
- C. -a > -d: This implies that d is less than 0 since the negation of a negative number is positive. However, we have no information about d, so we cannot confirm this statement.
- D. c + a < c - a: This states that adding a negative number to c is less than subtracting that same negative number (which is equivalent to adding the absolute value of a). This statement is always true regardless of c since adding a negative number (decreasing value) will always be less than adding a positive number (increasing value).
Hence, the correct answer is D. c + a < c - a, which must be true as adding a negative integer a will always give a smaller result than subtracting the negative integer a (which is equivalent to adding its positive value).