What happens to the inequality sign when you multiply or divide both sides by a negative number?

When you multiply or divide both sides of an inequality by a negative number, the inequality sign flips direction. This principle is crucial in maintaining the truth of the inequality. For instance, if you start with a true inequality such as (3 < 5) and you multiply both sides by (-1), the inequality becomes (-3 > -5). The sign change ensures that the relationship between the two sides remains valid.

This behavior is derived from how negative numbers are ordered on the number line, whereby multiplying or dividing by a negative number reverses the usual order. Thus, it's essential to remember this rule while solving inequalities to avoid errors that can lead to incorrect conclusions. Understanding this concept will greatly aid in solving inequality problems in algebra and further mathematical studies.