Why does the inequality sign change when both sides are multiplied or divided by a negative number? Does this happen with equations? Explain why or why not? Write an inequality for your classmates to solve. In your inequality, use both the multiplication and addition properties of inequalities.

Answer 1

take any true inequality, say ...

4 < 10
now multiply both sides by 10
40 < 100 , still true
divide original by 2
2 < 5 , still true

multiply original by -10
-40 < -100 , FALSE!, how do we make it true ?
-40 > -100

divide original by -2
-2 < -5 , FALSE! , how do we make it true?
-2 > -5

what about equations?
5 = 5 , true
multiply by -1
-5 = -5, (did you notice how I switched the = sign from = to = ?? , lol)

Answer 2

Another way to think of this changing direction stuff is to leave the direction the same, and move the expressions. Suppose you have

-3x > 9

Rather than dividing by -3, let's get rid of the negative coefficient by adding and subtracting.

Add 3x to both sides:

-3x + 3x > 9 + 3x
0 > 9 + 3x

Now subtarct 9 from both sides

0 - 9 > 9 + 3x - 9
-9 > 3x

Now divide by +3 to get

-3 > x

So, what we have is

-3x > 9 changes to
-3 > x

and even though we haven't change direction of the actual symbol, we have done it logically by moving things to the other side of the arrow.