iliavko, if you meant that to be an inequality, then yes, that's a valid move. (Of course you could do that with an equation, too.)
We only have to worry about flipping the sign when we actually multiply/divide by a negative. Let's take a look at an example:
x/y < 1
If y is positive, then we can just move it over and get x<y. For x/y to be less than 1, x would have to be either a smaller positive number (yielding a conventional fraction) or a negative number (yielding a negative, which of course will always be <1). For instance, x could be 2 and y could be 3, or x could be -2 and y could be any positive number.
If y is negative, we have to flip the sign when we move it over. x >y. Why is this so? Think about the kind of values you'd need to make it true. If we copy from our last example and just make the values negative, we could say x= -2 and y =-3. This of course will give us 2/3 and satisfy the inequality. But notice that while 2<3, -2 is actually GREATER THAN -3.
So basically, the whole reason we flip the sign is that negative numbers are counted backwards. The "smallest" ones are actually the biggest, making -1 much bigger than -100. Since the whole system runs opposite from the way we count positive numbers, whatever holds true on the positive side is reversed on the negative side.
For that reason, when we don't know if a variable is positive or negative, we can't multiply/divide it to the other side, because we don't know if we need to switch scenarios. No other action is going to cause the same problem (unless we're unsquaring or removing and absolute value sign), so the other moves we know are safe.