Implications of absolute value

Question

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

1. $|x+y|= x-y$
2. $|x+y| = -x+y$
3. $|x+y|=|x|-|y|$
4. $|x+y|=-(x+y)$

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?

Answer 1

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.