1/x<4. Multiply through by x: 1<4x, assuming x>0.
Divide through by 4: ¼<x, so x>¼, which is positive as assumed.
If x<0, then multiplying through by x reverses the inequality: 1>4x, ¼>x, x<¼, but this creates a contradiction because x can be less than ¼ but greater than zero, violating the condition x<0. Also x≠0. To resolve the contradiction, x<0.
Conclusion: x<0 or x>¼.
Check: Let x=½, then 1/x=2<4, so the inequality holds. Let x=1, then 1<4 and the inequality holds.
Let x=-¼, 1/x=-4<4 and the inequality holds. Let x=-1, -1<4 and the inequality holds.