On the interval $ [1, 10]$, use factoring and the triangle inequality to prove $|x-y|≤2\sqrt{10}|\sqrt{x} - \sqrt{y}|$.

Question

I need to show that on the interval [1, 10], $ |x-y|≤2\sqrt{10}|\sqrt{x} - \sqrt{y}|$ using factoring and the triangle inequality.

I know that I can factor the left side to $|(\sqrt{x}-\sqrt{y})(\sqrt{x}+\sqrt{y})|$ but for the life of me I can't figure it out beyond that. Any help would be appreciated.

Answer 1

$$\left|x-y\right|=\left|\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)\right|=\left|\sqrt{x}+\sqrt{y}\right|\left|\sqrt{x}-\sqrt{y}\right|$$ $$\le\left (\left|\sqrt{x}|+\left|\sqrt{y}\right|\right)\left|\sqrt{x}-\sqrt{y}\right|\le \left(|\sqrt{10}\right|+\left|\sqrt{10}\right|\right)\left|\sqrt{x}-\sqrt{y}\right|=2\sqrt{10}\left|\sqrt{x}-\sqrt{y}\right|.$$