Reciprocals and Inequalities

$$\frac{d}{dx} [ln(x)] = \frac{1}{x}$$

When to flip inequality signs

• Main Rule: When taking the reciprocal of an inequality (e.g., $a < b$), you must flip the inequality sign ($\frac{1}{a} > \frac{1}{b}$).

• Key Condition: This rule only works if both sides have the same sign.

Examples

• Both Positive:

  • $3 < 10 \implies \frac{1}{3} > \frac{1}{10}$

• Both Negative:

  • $-10 < -3 \implies -\frac{1}{10} > -\frac{1}{3}$

Exception

• If the signs are different, the inequality sign does not flip.
  • $-2 < 5 \implies -\frac{1}{2} < \frac{1}{5}$ (A negative is always less than a positive).