why it is necessary to reverse the inequality symbol when multiplying both sides of an inequality?
This only applies when multiplying or dividing the inequality by a negative number. The inequality has to change because once you have done the operation you are dealing with a whole new range of numbers (on the opposite side of zero). Another way to look at it (if one side is positive and the other is negative) is that the greater/less than sign has to continue to face whichever side (or sign, negative or positive) it faced in the beginning at all times.
