In fact, when we multiply or divide both sides of an inequality by *any negative
number,* the sign *always* flips. For instance, 4 > 2, so 4(- 3) < 2(- 3):
-12 < - 6. -2 < 6, so > : 1 > -3. This leads to the
multiplication and division properties of inequalities for negative numbers.

Multiplication and Division Properties of Inequalities for positive
numbers:

If *a* < *b* and *c* > 0, then *ac* < *bc* and <

If *a* > *b* and *c* > 0, then *ac* > *bc* and >

Multiplication and Division Properties of Inequalities for

**negative**
numbers:

If *a* < *b* and *c* < 0, then *ac* > *bc* and >

If *a* > *b* and *c* < 0, then *ac* < *bc* and <

**Note:** All the above properties apply to ≤ and ≥.

###
Properties of Reciprocals

Note the following properties:

If *a* > 0, then > 0

If *a* < 0, then < 0

When we take the reciprocal of both sides of an equation, something interesting happens--if the
numbers on both sides have the same sign, the inequality sign flips. For example,
2 < 3, but

> . Similarly,

> , but

-3 < . We can write this as a formal property:

If *a* > 0 and *b* > 0, or *a* < 0 and *b* < 0, and *a* < *b*, then >

If *a* > 0 and *b* > 0, or *a* < 0 and *b* < 0, and *a* > *b*, then <

**Note:** All the above properties apply to ≤ and ≥.