Inequality show that question involving two equations

[blopblop]

Hi, so here is my question that I am totally stumped on.

for all real values of x and y, show that |x|+|y|≥ √(x^2+y^2 )

and find the real values of x and y in which equality holds.


I sort of thought I could do the second part, but it confuses me with two pronumerals and how to get rid of one so that I can find the other.

 

[Vargo]

Both sides are positive so the inequality must hold if and only if the square of both sides satisfies the same inequality. That might simplify it a bit.

 

[kamke]

|x| + |y| ≥ √(x*x +y*y) / ( )2 squaring

x*x + y*y + 2*|x*y| ≥ x*x + y*y /-x*x,-y*y

2*|x*y| ≥ 0 /:2

|x*y| ≥ 0 It is true.


|x*y| = 0 ↔ x = 0 or y = 0

kamke