If I have 2x = 0, can someone explain what's going on or what rule doesn't allow for these two cases to exist...or can they and why...

Case 1:

2x = 0

(2x)/2 = 0/2

x = 0

O.K.

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Case 2:

2x = 0

x + x = 0

x = -x

x = (-1)x

x/x = ((-1)x)/x

1 = -1

???

...so if anybody could clear this up, it would be great! It's funny that after all of these years of doing math that I could stumble on something so basic that REALLY confuses me Thanks!