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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.