What is the effect of multiplying or dividing an inequality by a negative number?

When multiplying or dividing an inequality by a negative number, it is essential to switch the inequality sign. This rule stems from the behavior of numbers on the number line. For instance, if you take the inequality ( x < y ) and multiply both sides by a negative number, say (-1), the relationship between the two numbers reverses. This means that what was once ( x < y ) would then become ( -x > -y ).

This reversal occurs because the order of numbers changes when multiplied or divided by a negative number. Such a change is necessary to maintain the truth of the inequality. Therefore, switching the inequality sign is crucial to correctly represent the relationship after such a manipulation. This is a foundational property of inequalities and is a critical concept in algebra, ensuring accuracy when solving problems that involve inequalities.