Thread: need help understanding this absolute value proof?

Prove that for every two real numbers x and y, |x+y|>=|x|-|y|. Hint observe that |x|=|(x+y|+(-y)|. I get no idea how they they get that observation. is it because |x+y|>=|x|+|y|= |x+y|-|y|>=|x|? but then not sure where to go from here?

I believe the inequality sign should be reversed, i.e. .

Here's a hint: "Triangle inequality"

Oh my bad it was supposed to be |x+y|>=|x|-|y|. I think I get how they get that. But it makes sense though because in the end you |x|=|x| from that observation. But by using the observation that was given here is the proof i came up with. Since |x+y|>=|x|-|y| which means that |x|=|(x+y+(-y)|. By the triangle inequality this becomes |(x+y)+(-y)|>=|x+y|+|-y|=|x+y|+|y|>= |x+y|>=|x|-|y|. Hence |x+y|>=|x|-|y|