To determine which of the expressions are equivalent to [tex]\(\frac{p}{-q}\)[/tex], let's analyze each option individually.
### Option A: [tex]\(-\frac{-p}{-q}\)[/tex]
First, let's simplify the expression:
1. [tex]\(-\frac{-p}{-q}\)[/tex]
2. Simplify the negative signs in the numerator and the denominator:
3. [tex]\(-\left(\frac{-p}{-q}\right) = -\left(\frac{p}{q}\right) = -\frac{p}{q}\)[/tex]
This simplifies to [tex]\(-\frac{p}{q}\)[/tex], which is not equivalent to [tex]\(\frac{p}{-q}\)[/tex]. Therefore, Option A is not equivalent to [tex]\(\frac{p}{-q}\)[/tex].
### Option B: [tex]\(\frac{-p}{q}\)[/tex]
Let's simplify this expression:
1. [tex]\(\frac{-p}{q}\)[/tex]
This represents a fraction where both the numerator and the denominator are divided as usual, retaining the negative sign in the numerator. Hence, [tex]\(\frac{-p}{q}\)[/tex] is indeed equivalent to [tex]\(\frac{p}{-q}\)[/tex]. Therefore, Option B is equivalent to [tex]\(\frac{p}{-q}\)[/tex].
### Option C: None of the above
Given that Option B is indeed equivalent to [tex]\(\frac{p}{-q}\)[/tex], Option C cannot be correct since it suggests that none of the given options are equivalent.
### Conclusion
After this detailed analysis, we determine that:
- Option A: [tex]\(-\frac{-p}{-q}\)[/tex] is not equivalent to [tex]\(\frac{p}{-q}\)[/tex].
- Option B: [tex]\(\frac{-p}{q}\)[/tex] is equivalent to [tex]\(\frac{p}{-q}\)[/tex].
- Option C: "None of the above" is not correct since Option B is equivalent to [tex]\(\frac{p}{-q}\)[/tex].
Thus, the correct answers are:
- Option A: 0 (not equivalent)
- Option B: 1 (equivalent)
- Option C: 0 (not applicable)
