Adding/Subtracting Negative Numbers

What I'm wondering is if when you make the numbers in the problem their absolute values, why does changing the sign work as well? Is it a property of absolute values that when both numbers become positive, you always change the sign? Or do you NOT always change the sign? Does making numbers in the problem their absolute value ALWAYS mean the sign in between them is - ? If so then this next problem doesn't make sense that you would ALWAYS use the sign of the bigger number as well, because the answer is negative. Does that mean that if the sign is already negative when you make them absolute values that you ALWAYS use the sign of the smaller number? I can do these problems in my head, but i want to be able to understand exactly why. Sorry I know i sound goofy asking this. Thank you guys SO much for helping with this.

ex. -4 - 7 =

is it |4| - |7| = -3 (i know this isn't right)

or |4| + |7| = 11

This isn't right either because its positive when it should be -11 . In these kinds of problems (when you change from subtracting to adding by changing the sign along with absolute values) do you always use the sign of the smaller number instead of the bigger one? So if the sign starts out as a + you use the sign of the larger number, but if the sign starts out as a - you use the sign of the smaller number? If so, does this rule ALWAYS apply?

Re: Adding/Subtracting Negative Numbers

your both the statements are correct.

in second case |4|-|7| = -3 because you are subtracting the absolute values.

In second case you are adding the absolute values. So there should be no confusion. -4-7=-11

Just remember that the absolute value or modulus returns positive value.

That is |-5| = 5; |-x| = x ; |a| = a etc.

Thus |-a| + |b| = a+b; |-a| - |b| = a-b; |-a| - |-b| = a - b etc