In solving inequalities, what must you remember when multiplying or dividing by a negative number?

When solving inequalities, one crucial aspect to remember is that if you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. This rule is rooted in the properties of inequalities and is essential to maintaining the correct relationship between the two sides.

For example, consider the inequality ( 3 < 5 ). If we multiply both sides by -1, the inequality becomes ( -3 ) and ( -5 ). However, if we do not reverse the sign and simply write ( -3 < -5 ), it is incorrect, as -3 is actually greater than -5. Thus, by reversing the inequality sign, it correctly reflects the new relationship: ( -3 > -5 ).

This rule ensures that the inequality accurately represents the values and their relationships after the operation, upholding the truth of the statement through manipulation. Understanding this principle helps prevent errors when working with inequalities involving multiplication or division by negative numbers.