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## Homework Statement

Verify the rule that for two real numbers X and Y then |X+Y|≤|X|+|Y|

## Homework Equations

## The Attempt at a Solution

1. When all the variables are positive:

|X+Y|=(X+Y) because (X+Y)>0

|X|=X because X>0

|Y|=Y because Y>0

So we got

(X+Y)=X+Y

2. When all the variables are negative:

|-X+(-Y)|= -(-[X+Y])= (X+Y) because -(X+Y)<0

|-X|= -(-X)=X because -X<0

|-Y|= -(-Y)=Y because -Y<0

Now we got

(X+Y)=X+Y

3. When the two variables have opposite signs:

|X+(-Y)|= (X-Y) because (X-Y)>0

|X|=X because X>0

|-Y|= -(-Y)=Y because Y<0

Finally we got:

(X-Y)< X+Y

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So that solution was shot down as a bit shaky. How about the one below then?

|X+Y|≤|X|+|Y|

|X+Y|-|X|-|Y|≤0

If we remove the variables from the modulus we get

X+Y-X-Y≤0

Thanks.