Rules for solving inequalities with negative fractions

Question

What are the algebraic rules for solving inequalities with negative signs and fractions like:

$$\frac{1}{x}<-\frac{1}{5}$$

Answer 1

1. Whenever we take the inverse, and both sides are either positive or both sides are negative then the inequality sign reverses; (if one is side is positive and one side is negative, then the sign stays the same.)

2. Whenever we multiply both sides by $-1$, the inequality sign reverses too.

Since you don´t know whether x is positive or negative, you need to consider both scenarios.

Answer 2

If $x \in \mathbb{R}$, then $\frac{1}{x} < -\frac{1}{5}$ iff $x > -5$ and iff $-x < 5$.

If you are seeking after the rule of thumb, here it is: whenever we take inverse of both sides, the inequality sign reverses; whenever we multiply both sides by $-1$, the inequality sign reverses too.