Why do you not change the inequality sign when you are adding or subtracting?
Because to do so would be algebraically incorrect. See below.
Consider the simplest of inequalities:
Now consider adding or subtracting a real number,
The only way to restore the inequality is to add or subtract
So when must we reverse the inequality?
Consider where we multiply (or divide) both sides of the inequality by
As an example I will use
So, in order to maintain the inequality after multiplying or dividing through by a negative number we must reverse the inequality.
Hope this helps. It's not as complicated as it seems!