Why does the inequality sign change when both sides are multiplied or divided by a negative number? Does this happen with equations? Why or why not?
Numbers are directed quantities, if nothing else, they are directed on the number line from (conventionally, left side, negative to positive. That is what the > sign means, it means more to the right).
-1 is a directed operation: it means NOT 1., the reverse direction from 1.
So multiplying both sides of an equation means a NOT operation, or reverse direction.
a<b
-a>-b