SOLUTION: why does the inequality sign change when both sides are multiplied or divided by a negative number? does this happen with equations?

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Question 393408: why does the inequality sign change when both sides are multiplied or divided by a negative number? does this happen with equations?
Found 2 solutions by solver91311, Edwin McCravy:
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


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
The Out Campaign: Scarlet Letter of Atheism

Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
why does the inequality sign change when both sides are multiplied or divided by a negative number? does this happen with equations?


Suppose we have this inequality

   X  <  Y

Suppose -A is a negative number, which means A is a positive number.

We want to prove that

  -AX > -AY

----------------------

Start with

           X  <  Y

Multiply both sides by A, a positive number.
That does not change the inequality sign:

          AX  <  AY 

Add -AX - AY to both sides, which also does
not change the inequality sign:

 AX - AX - AY < AY - AX - AY

          -AY < -AX

But that is the same as

          -AX > -AY

which is what we wanted to prove.

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Does this happen with equations?

Yes it does but if you reverse an = sign, it looks 
exactly the same! Haha! J

Edwin