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.