Why Does the Inequality Sign Change When Both Sides Are Multiplied or Divided by a Negative Number?
Multiplying or dividing both sides of an equation by a negative number changes the inequality of the equation, because it changes the sign of each side of the equation. On both sides, what was positive becomes negative, and what was negative becomes positive.
Positive numbers are greater than zero, and are denoted with a + sign or no sign at all. Negative numbers are less than zero, and they're denoted with a - sign. As positive numbers increase in magnitude, they become larger. For instance, 3 is greater than 2, which is greater than 1. As negative numbers increase in magnitude, they become smaller. Therefore, -3 is smaller than -2, and -2 is smaller than -1.
An inequality equation compares the values of both sides of an equation. When both sides of an inequality equation reverse signs because they've been multiplied or divided by a negative number, each side's comparative value to the equation's other side also reverses. This concept can be challenging to grasp when the equation has unknown variables, such as x and y. Illustrating this principle with a simple inequality equation with known numbers makes things clearer.
Take a simple mathematical statement like 7 > 3. This presents the mathematical relationship between 7 and 3 as 7 is greater than 3. If both sides of the equation are multiplied or divided by -1, the left side of the equation becomes -7, and the right side becomes -3. Now that the numbers are negative, -7 is not greater than -3; it is smaller. The statement -7 > -3 is wrong. Therefore, the inequality sign reverses to make the statement correct: -7 < -3.