Let's analyze the given statements one by one, using the properties of negative rational numbers:
1. Statement A: [tex]\(x \cdot y \cdot z\)[/tex] is positive
- When you multiply three negative numbers, the result is positive. Multiplying two negative numbers gives a positive result and then multiplying that positive result by another negative number gives a negative result. Therefore, Statement A is incorrect because [tex]\(x \cdot y \cdot z\)[/tex] is not positive when [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex] are negative rational numbers.
2. Statement B: [tex]\(x - y - z\)[/tex] is negative
- Let's explore this: [tex]\(x - y - z\)[/tex]. Given that [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex] are negative, let's assume [tex]\(x = -a\)[/tex], [tex]\(y = -b\)[/tex], and [tex]\(z = -c\)[/tex], where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are positive rational numbers.
- Substituting these:
[tex]\[
x - y - z = -a - (-b) - (-c) = -a + b + c
\][/tex]
- Since [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are positive, the expression [tex]\(-a + b + c\)[/tex] can be either negative or positive, depending on the relative sizes of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]. Thus, we cannot definitively say that [tex]\(x - y - z\)[/tex] is always negative. Therefore, Statement B is not necessarily true.
3. Statement C: [tex]\(x + y + z\)[/tex] is positive
- For the sum [tex]\(x + y + z\)[/tex], since [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex] are all negative rational numbers, the sum of any three negative numbers is also negative. Therefore, Statement C must be true. This is the correct statement.
4. Statement D: [tex]\(x \cdot y \cdot z\)[/tex] is negative
- As discussed in Statement A, multiplying three negative numbers results in a negative product. Therefore, [tex]\(x \cdot y \cdot z\)[/tex] is indeed negative. Since this statement contradicts Statement A but does not contradict Statement C, we might need to re-evaluate.
After a thorough review, the conclusive statement is:
Out of the given options, the statement that always holds true if [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex] are negative rational numbers is:
C) [tex]\(x + y + z\)[/tex] is always negative.
