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

Question 1

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?

Question 2

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 $(-\infty,0)\cup(1/4, \infty).$

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

Answer

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