Question

Find the set of all solutions of the equation
$2^{|y|}-|2^{y-1}-1|=2^{y-1}+1$

Answer 1

Given, $2^{|y|}-|2^{y-1}-1|=2^{y-1}+1$
Case I When $y\in(-\infty,0]$
$\therefore 2^{-y}+(2^{y-1}-1)=2^{y-1}+1$
$\Rightarrow 2^{-y}=2$
$\Rightarrow y=-1\in(-\infty,0]...(i)$
Case II When $y\in(0,1]$
$\therefore 2^y+(2^{y-1}-1)=2^{y-1}+1$
$\Rightarrow 2^y=2$
$\Rightarrow y=1\in(0,1] ...(ii)$
Case III When $y\in(1,\infty)$
$\therefore 2^y-2^{y-1}+1=2^{y-1}+1$
$\Rightarrow 2^y-2.2^{y-1}=0$
$\Rightarrow 2^y-2^y=0$ true for all $y>1 ...(iii)$
From Eqs. (i), (ii) and (iii), we get
$y\in\big\{-1\big\}\cup[1,\infty\big).$