Look at an example:
We know that 2 is most certainly less than 3, hence the statement
is most certainly a true statement. But if you multiply by -1:
Of course not. -2 is certainly larger than -3. Hence you have to change the sense of the inequality.
Any positive number gets larger as the absolute value of that positive number gets larger. Any negative number gets SMALLER as the absolute value of that number gets LARGER. Hence, negative numbers have an opposite sense from positive numbers and you have to change the inequality sense when multiplying by a negative number.
As to the second part of your question, there are two equally valid (in my mind) theories. One theory says "No, the equals sign is not reversed when you multiply an equation by a negative number." The other theory says, "Yes, the equals sign IS reversed when you multiply by a negative number, it is just that you can't tell which way any given equals sign is facing." Take your pick.
John
My calculator said it, I believe it, that settles it
