Discrete Mathematics Absolute Value Proof

[tennesseewiz]

Homework Statement

Prove the following statement:
For all real numbers x and y, |x| times |y| = |xy|

Homework Equations

I really don't know how to start this as a formal proof.

The Attempt at a Solution

I was thinking I'd have to break it down into four cases and logically prove that the statement is true because no matter what, x times y is going to have the same numerical value as it's opposite number (of course beside it being negative) because once you take the absolute value, it's going to be positive anyways.
Case 1: Suppose both x and y are positive real numbers.
Case 2: Suppose x is a negative real number and y is a positive real number.
Case 3: Suppose x is a positive real number and y is a negative real number.
Case 4: Suppose both x and y are positive.

Am I on the right track or am I going in the wrong direction?

 

[JonF]

if you want to do cases you only need to do 3. You can WLOG two of them together.

 

[tennesseewiz]

Do you mean cases 2 and 3 then?

 

[HallsofIvy]

Yes, the situation where a> 0 and b< 0 is exactly the same as a< 0 and b>0. However, I would not discourage you from considering the two cases separately. You are completely correct to argue that there are 2 cases for x and 2 cases for y and so (2)(2)= 4 cases altogether.