Question Number 193080 by Mastermind last updated on 03/Jun/23
$$\mathrm{Show}\:\mathrm{that}\:\mathrm{for}\:\mathrm{a},\mathrm{b}\in\mathbb{R}, \\ $$$$\mid\mathrm{a}−\mathrm{b}\mid\geqslant\mid\mathrm{a}\mid−\mid\mathrm{b}\mid \\ $$$$ \\ $$$$\left(\mathrm{Hint}:\:\mathrm{write}\:\mathrm{a}=\left(\mathrm{a}−\mathrm{b}\right)+\mathrm{b}\right. \\ $$
Answered by Subhi last updated on 03/Jun/23
$$\left.\mid{a}\mid−\mid{b}\mid\:=\:\mid\left({a}−{b}\right)+{b}\right)\mid−\mid{b}\mid \\ $$$${note}\:{that}:\:\mid{a}+{b}\mid\leqslant\mid{a}\mid+\mid{b}\mid \\ $$$$\mid\left({a}−{b}\right)+{b}\mid−\mid{b}\mid\leqslant\mid\left({a}−{b}\right)\mid+\mid{b}\mid−\mid{b}\mid=\mid{a}−{b}\mid \\ $$$$\therefore\:\mid{a}−{b}\mid\geqslant\mid{a}\mid−\mid{b}\mid \\ $$