What is the name of the inequality $|x|+|y| \geq |x+y|$?

I remember seeing this inequality and thinking it was the triangle inequality, but that only holds if $x,y,z$ are the side lengths of a triangle and this holds for all real numbers? If so is there a name for this inequality?

  • 5
    $\begingroup$ Still called the "triangle" inequality. $\endgroup$ Feb 7, 2016 at 15:50
  • $\begingroup$ Why is it called that? $\endgroup$
    – Puzzled417
    Feb 7, 2016 at 16:00

1 Answer 1


If $\bar{x}$, $\bar{y}$ are two vectors, $\bar{x}+\bar{y}$ is the vector representing the third side of the triangle when the tip of $\bar{x}$ touches the tail of $\bar{y}$.

(Look at the triangle law of adding vectors to get a sense of the respective directionalities).

The inequality translates as "Sum of lengths of two sides of a triangle is greater than length of the third side".


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