Implications of absolute value Does the following given conjunction $$ x\lt 0\le y \wedge |x|\gt |y|$$ implies that either


*

*$|x+y|= x-y$ 

*$|x+y| = -x+y$

*$|x+y|=|x|-|y|$

*$|x+y|=-(x+y)$


or a combination of those proposition, is true?
I look at absolute values as being the magnitudes of vectors so geometrically the result that makes sense to me would be to say that $1$, $2$ are false and $3$, $4$ are true such that
$$ x\lt 0\le y \wedge |x|\gt |y| \implies |x+y|=|x|-|y|\wedge  \;|x+y|=-(x+y)\ $$
would be the correct implication. Is this the only true implication?
 A: Your geometric intuition is good. Here is an arithmetic proof.
For convenience (to avoid having to write the same thing many times),
let $P$ be the conjunction 
$x < 0 \leq y \land \lvert x \rvert > \lvert y \rvert.$
Suppose $P$ is true. Then:
Since $x < 0 \leq y,$ we know that 
$\lvert x \rvert = -x$ and $\lvert y \rvert = y.$
Since $\lvert x \rvert > \lvert y \rvert,$
we have $y < -x,$ so $x + y < 0$ and 
$\lvert x + y \rvert = -(x+y).$ This proves part 4.
Furthermore, $-(x+y) = -x - y = \lvert x \rvert - \lvert y \rvert,$
so $\lvert x + y \rvert = \lvert x \rvert - \lvert y \rvert.$
This proves part 3.
On the other hand, 
$\lvert x + y \rvert - (x - y) = -(x + y) - (x - y) = -2x > 0,$
so $\lvert x + y \rvert \neq (x - y),$
disproving part 1. (That is, $P$ implies statement 1 is false.)
Also, $\lvert x + y \rvert - (-x + y) = -(x + y) - (-x + y) = -2y.$
That is, $\lvert x + y \rvert = (-x + y) - 2y.$
Therefore $P$ implies that statement 2 is true if and only if $y = 0$;
$P$ alone (without the additional condition $y=0$)
do not imply statement 2.
So the conjunction $x < 0 \leq y \land \lvert x \rvert > \lvert y \rvert$
implies statements 3 and 4 but not statements 1 and 2.
