Question

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

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

Answer 1

Answer: (D)

$\left(-\infty , 0\right)\cup\left(1, \infty\right)$

Answer 2

$1-e^{\frac{1}{x}-1} >0$ $\Rightarrow e^{\frac{1}{x}-1} < 1$ $\Rightarrow \frac{1}{x}-1 < log\,1$ $\Rightarrow \frac{1}{x}-1 < 0$ $\Rightarrow \frac{1}{x} < 1$ $\Rightarrow x\,\in\left(-\infty, 0\right)\cup\left(1, \infty\right)$