Question

If D is the set of all real x such that \( 1 - e^{(1/x)} \) is positive, then D is equal to:

• A. \( (-\infty, -1] \)
• B. \( (-\infty, 0) \)
• C. \( (1, \infty) \)
• D. \( (-\infty, 0) \cup (1, \infty) \)

Hint:

Solve inequalities involving exponentials carefully, keeping in mind the behavior of exponential functions for positive and negative values of \( x \).

Answer 1

The Correct Option is D

Step 1: Solve the inequality.
For the inequality \( 1 - e^{(1/x)}>0 \), solve for \( x \). The condition implies that \( x \) must be in the union of the intervals \( (-\infty, 0) \cup (1, \infty) \).
Step 2: Conclusion.
Thus, the set D is \( (-\infty, 0) \cup (1, \infty) \).
Final Answer: \[ \boxed{(-\infty, 0) \cup (1, \infty)} \]