To determine whether the expression \((-x)/((-y)(-z))\) is positive, negative, or cannot be determined given that \(xyz > 0\), we can follow these steps:
### Step 1: Analyze the given condition
We know that \(xyz > 0\). This means that the product of \(x\), \(y\), and \(z\) is positive. This can occur in two scenarios:
1. All three variables \(x\), \(y\), and \(z\) are positive.
2. One variable is negative, and the other two are negative (since the product of two negative numbers is positive).
### Step 2: Rewrite the expression
The expression can be rewritten as:
\[
\frac{-x}{(-y)(-z)} = \frac{-x}{yz}
\]
This is because multiplying two negative numbers results in a positive number.
### Step 3: Determine the sign of the expression
Now, we need to analyze the sign of \(-x\) and \(yz\):
- The term \(yz\) is positive since \(xyz > 0\) and \(x\) is either positive or negative.
- The term \(-x\) will be negative if \(x\) is positive, and it will be positive if \(x\) is negative.
### Step 4: Consider the cases for \(x\)
1. **Case 1:** If \(x > 0\) (meaning \(y\) and \(z\) must also be positive to keep \(xyz > 0\)):
- Then, \(-x < 0\) and \(yz > 0\).
- Therefore, \(\frac{-x}{yz} < 0\) (the expression is negative).
2. **Case 2:** If \(x < 0\) (meaning \(y\) and \(z\) must both be negative to keep \(xyz > 0\)):
- Then, \(-x > 0\) and \(yz > 0\).
- Therefore, \(\frac{-x}{yz} > 0\) (the expression is positive).
### Conclusion
Based on the analysis:
- If \(x > 0\), the expression is negative.
- If \(x < 0\), the expression is positive.
Thus, the expression \((-x)/((-y)(-z))\) cannot be determined without knowing the sign of \(x\).
### Numerical Examples
To illustrate the conclusion:
- **Example 1 (Negative Expression):** Let \(x = 1\), \(y = 2\), \(z = 3\). Then \(xyz = 1 \cdot 2 \cdot 3 = 6 > 0\) and \(\frac{-1}{(-2)(-3)} = \frac{-1}{6} < 0\).
- **Example 2 (Positive Expression):** Let \(x = -1\), \(y = -2\), \(z = 3\). Then \(xyz = -1 \cdot -2 \cdot 3 = 6 > 0\) and \(\frac{-(-1)}{(-(-2))(-3)} = \frac{1}{6} > 0\).
### Final Answer
The expression \((-x)/((-y)(-z))\) is **Cannot Be Determined**.
