Question

Cho \(x,y\in Q\). Chứng minh \(\left|x-y\right|\ge\left|x\right|-\left|y\right|\)

Answer 1

\(\left|x\right|=\left|\left(x-y\right)+y\right|\le\left|x-y\right|+\left|y\right|\\ \Rightarrow\left|x\right|-\left|y\right|\le\left|x-y\right|\)

Dấu \("="\Leftrightarrow xy\ge0\)

Answer 2

\(\left|x-y\right|\ge\left|x\right|-\left|y\right|\)

⇒ \(\left(\left|x-y\right|\right)^2\ge\left(\left|x\right|-\left|y\right|\right)^2\)

⇒ \(\left(x-y\right)^2\ge x^2+2\left|xy\right|-y^2\)

⇒ \(x^2-2xy-y^2\ge x^2-2\left|xy\right|-y^2\)

⇒ 2xy \(\ge\) \(2\left|xy\right|\)

Kết luận: ...

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