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

Solving 1x<4 gives me x>14. The book however states the answer is: x<0 or x>14.

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?

Answer

You have to be careful when multiplying by x since x might be negative and hence flip the inequality. Suppose x>0. Then
1x<44x>1x>1/4.
If x>0 and x>1/4, then x>1/4.

Now suppose x<0. Then
1x<44x<1x<1/4.
If x<0 and x<1/4, then x<0.
So the solution set is (,0)(1/4,).

Attribution
Source : Link , Question Author : Augusto Dias Noronha , Answer Author : Sri-Amirthan Theivendran

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