Of 0<st<1, which of the following can be true?
(a). s<-1 and t>0
(b). s<-1 and t<-1
(c). s>-1 and t<-1
(d). s>1 and t<-1
(e). 1/x<0<1/y

Given the condition \(0 < s t < 1\), we need to analyze each option to see which one can be true under this constraint.

First, consider the constraint \(0 < s t < 1\):
- This implies that \(s\) and \(t\) must be such that their product \(st\) is positive and less than 1.
- Therefore, both \(s\) and \(t\) must either both be positive or both be negative.

Let's analyze each option one by one:

(a) \(s < -1\) and \(t > 0\):
- If \(s < -1\) and \(t > 0\), then \(s t < -t\), and since \(t > 0\), \(s t\) will be negative. This contradicts \(0 < s t\). So, this cannot be true.

(b) \(s < -1\) and \(t < -1\):
- If \(s < -1\) and \(t < -1\), then \(s t > 1\) since both values are negative and their product will be positive but greater than 1. This contradicts \(0 < s t < 1\). So, this cannot be true.

(c) \(s > -1\) and \(t < -1\):
- If \(s > -1\) and \(t < -1\), \(s\) must be negative for \(s t\) to be positive, since \(t\) is negative. Let's choose \(s\) such that \(-1 < s < 0\). Then \(s t\) will be positive and if \(s\) is sufficiently close to 0, \(s t\) can be less than 1. For example, if \(s = -0.5\) and \(t = -2\), \(s t = 1\), but to satisfy the inequality strictly, we need \(s t < 1\), so adjusting values slightly can make this true. Therefore, this can be true.

(d) \(s > 1\) and \(t < -1\):
- If \(s > 1\) and \(t < -1\), then \(s t < -s\), and since \(s > 1\), \(s t\) will be negative. This contradicts \(0 < s t\). So, this cannot be true.

(e) \(\frac{1}{x} < 0 < \frac{1}{y}\):
- For \(\frac{1}{x} < 0\), \(x\) must be negative. For \(\frac{1}{y} > 0\), \(y\) must be positive. However, this condition does not directly relate to the given constraint \(0 < s t < 1\), so it is irrelevant in this context.

Therefore, the correct answer is (c) \(s > -1\) and \(t < -1\).

Would you like any further details or have any questions? Here are 5 questions you might consider asking next:
1. How can I determine the range of a function?
2. What is the process to solve a quadratic equation?
3. Can you explain the concept of limits in calculus?
4. How do I find the derivative of a given function?
5. What are the steps to solve a system of linear equations?

**Tip:** When analyzing inequalities involving products, consider the signs of the factors carefully to determine if the product can satisfy the given conditions.

Explore the analysis of the inequality 0 < st < 1 and determine which statements can be true. Understand the conditions where the product of two variables s and t is strictly between 0 and 1. Suitable for advanced high school students and those interested in algebraic inequalities.

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