How do I justify this absolute value problem?

Question

Is the following statement true for all x, y ∈ R? Justify your answer. |x| + |y| = |x + y|

Answer 1

It is not always true.  If x and y have opposite signs, the equality is not true.  Example:

 

|-2| + |5| = 2 + 5 = 7

|-2 + 5| = 3 ≠ 7

 

However, if x and y have the same sign (both positive or both negative), the equality is true.  Examples:

 

Both positive:

 

|2| + |3| = 5

|2 + 3| = 5

 

Both negative:

 

|-2| + |-3| = 5
|-2 + -3| = 5

 

Similarly, if either one is positive or negative and the other is zero, the equality is true.