# Why does 1x<4\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?

You have to be careful when multiplying by $x$ since $x$ might be negative and hence flip the inequality. Suppose $x>0$. Then
If $x>0$ and $x>1/4$, then $x>1/4$.
Now suppose $x<0$. Then
If $x<0$ and $x<1/4$, then $x<0$.
So the solution set is $(-\infty,0)\cup(1/4, \infty).$