Problem Set - Absolute Value

## Problem Set - Absolute Value

 Let $x\in ℝ$. Define $|x|$. Let $x\in ℂ$. Define $|x|$. Let $x\in ℝ$. Show that $|x|=|x+0i|$. Let $x\in ℝ$. Show that $|-x|=|x|$. Let $x,y\in ℝ$. Show that $|x+y|\le |x|+|y|$. Let $x,y\in ℂ$. Show that $|x+y|\le |x|+|y|$. Let $x,y,z\in ℝ$. Show that $|x+y+z|\le |x|+|y|+|z|$. Let $x,y,z\in ℂ$. Show that $|x+y+z|\le |x|+|y|+|z|$. Let $x,y\in ℂ$. Show that ${|x+y|}^{2}+{|x-y|}^{2}=2\left({|x|}^{2}+{|y|}^{2}\right)$. Let $x,y\in ℂ$. Show that ${|x+y|}^{2}={|x|}^{2}+{|y|}^{2}+2\mathrm{Re}\left(a\overline{b}\right)$. Let $x,y\in ℝ$. Show that $|x+y|\ge |\text{\hspace{0.17em}}|x|-|y|\text{\hspace{0.17em}}|$. Let $x,y\in ℝ$. Show that $|x-y|\ge |\text{\hspace{0.17em}}|x|-|y|\text{\hspace{0.17em}}|$. Let $x,y,z\in ℝ$. Show that $|x+y+z|\ge |\text{\hspace{0.17em}}|x|-|y|-|z|\text{\hspace{0.17em}}|$. Give solutions to the following inequalities in terms of intervals: (a)   $|x|>3$. (b)   $|1+2x|\le 4$. (c)   $|x+2|\ge 5$. (d)   $|x-5|<|x+1|$. (e)   $|x-2|<3$ or $|x+1|<1$. (f)   $|x-2|<3$ and $|x+1|<1$. Let $a,b\in ℝ$ and let $0<\epsilon <|b|$. Show that $|\frac{a+\epsilon }{b+\epsilon }|\le \frac{|a|+\epsilon }{|b|+\epsilon }$. Prove that if ${a}_{1},{a}_{2},\dots ,{a}_{n}\in ℝ$ then $|\sum _{k=1}^{n}{a}_{k}|\le \sum _{k=1}^{n}|{a}_{k}|$. Prove that if ${a}_{1},{a}_{2},\dots ,{a}_{n}\in ℝ$ then $|\sum _{k=1}^{n}{a}_{k}|\ge |{a}_{p}|-\sum _{k=1,k\ne p}^{n}|{a}_{k}|$.