1/x < 4 solve the inequality
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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.

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