Ta có: \(\left|x\right|\ge x;\left|x\right|\ge-x\forall x\)
\(\left|y\right|\ge y;\left|y\right|\ge-y\forall y\)
\(\Rightarrow\left|x\right|+\left|y\right|\ge x+y;\left|x\right|+\left|y\right|\ge-\left(x+y\right)\)
\(\Rightarrow x+y\ge-\left(\left|x\right|+\left|y\right|\right)\)
Do đó, \(-\left(\left|x\right|+\left|y\right|\right)\le x+y\le\left|x\right|+\left|y\right|\)
\(\Rightarrow\left|x+y\right|\le\left|x\right|+\left|y\right|\)
Dấu ''='' xảy ra khi \(xy\ge0\)
\(\left|x+y\right|\le\left|x\right|+\left|y\right|\)
\(\Leftrightarrow\left(\left|x+y\right|\right)^2\le\left(\left|x\right|+\left|y\right|\right)^2\)
\(\Leftrightarrow x^2+2xy+y^2\le x^2+2\left|xy\right|+y^2\)
\(\Leftrightarrow2xy\le2\left|xy\right|\)
\(\Leftrightarrow xy\le\left|xy\right|\) (luôn đúng)
Dấu = khi \(xy\ge0\)
