Why does $\frac{1}{x} < 4$ have two answers?

You have to be careful when multiplying by $x$ since $x$ might be negative and hence flip the inequality. Suppose $x>0$. Then $$\frac{1}{x}<4\iff4x>1\iff x>1/4.$$ If $x>0$ and $x>1/4$, then $x>1/4$.

Now suppose $x<0$. Then $$\frac{1}{x}<4\iff4x<1\iff x<1/4.$$ If $x<0$ and $x<1/4$, then $x<0$. So the solution set is $(-\infty,0)\cup(1/4, \infty).$

Here is the solution $$\frac { 1 }{ x } <4$$$$ \frac { 1-4x }{ x } <0$$$$ \frac { x\left( 1-4x \right) }{ { x }^{ 2 } } <0$$$$ x\left( 1-4x \right) <0$$$$ x\left( 4x-1 \right) >0 $$

so $$x\in \left( -\infty ,0 \right) \cup \left( \frac { 1 }{ 4 } ,+\infty \right) $$

I am following the suggestion given by @quid in the comments because I like pictures:

The orange/red line is the $x$-axis. The yellow line is the line $y=4$. The two blue curves are the graph of $y=1/x$. The solution to the inequality is the set of $x$ values for which the blue curve is below the yellow line. As @quid predicted, this picture gives some intuition for why there are two "zones" in the solution.

Note: originally I had $y=\frac{1}{4}$ which was incorrect, so I changed the answer to reflect the fact that it should be $y=4$ and re-plotted the graph.