Why does $\frac{1}{x} < 4$ have two answers? Solving $\frac{1}{x} &lt; 4$ gives me $x &gt; \frac{1}{4}$. The book however states the answer is: $x &lt; 0$ or $x &gt; \frac{1}{4}$.

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 $1 &lt; 4x$ 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 $x&gt;0$. Then
$$\frac{1}{x}&lt;4\iff4x&gt;1\iff x&gt;1/4.$$
If $x&gt;0$ and $x&gt;1/4$, then $x&gt;1/4$.

Now suppose $x&lt;0$. Then 
$$\frac{1}{x}&lt;4\iff4x&lt;1\iff x&lt;1/4.$$
If $x&lt;0$ and $x&lt;1/4$, then $x&lt;0$.
So the solution set is $(-\infty,0)\cup(1/4, \infty).$
$$\frac{1}{x} &lt; 4,$$
$$\frac{1}{x} - 4 &lt; 0,$$
$$\frac{1 - 4x}{x} &lt; 0,$$
$1 - 4x &lt; 0$ or $x &gt; 0$. Because if $1 - 4x$ is negative then $x$ must be positive. So we must write the solution in that way.