Why does $\frac{1}{x} < 4$ have two answers? Solving $\frac{1}{x} < 4$ gives me $x > \frac{1}{4}$. The book however states the answer is: $x < 0$ or $x > \frac{1}{4}$.
My questions are:
Why does this inequality has two answers (preferably the intuition behind it)?
When using Wolfram Alpha it gives me two answers, but when using $1 < 4x$ it only gives me one answer. Aren't the two forms equivalent?

I would like to harp on the meaning of your "two answers". When a question asking for a number satisfying some conditions leads to a unique number having that property we are fine. When there are two numbers having the required property then we can say two answers.
Here we have uncountably many real numbers $x$ such that $\frac1x < 4$. So even the region $x>4$ is infinitely many answers. One possible interpretation that could justify "two" could be number of connected components of the solution space of the question.  This happens many times. The set $GL(n,\mathbf{R})$ of non-singular real $n\times n$ matrices has two connected components.
