Question

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

Answer 1

When $2^{y-1}-1<0$,

$2^{y-1}<1\Rightarrow 2^{y-1}<2^0\Rightarrow y-1<0\Rightarrow y<1$ ($\because 2^x$ is a strictly increasing function)

Case 1: ($y<0$)

$\Rightarrow 2^{(-y)}-(1-2^{y-1})=2^{y-1}+1\Rightarrow 2^{(-y)}=2\Rightarrow y=(-1)$

Case 2: ($0\le y<1$)

$\Rightarrow 2^y-(1-2^{y-1})=2^{y-1}+1\Rightarrow 2^y=2\Rightarrow y=1$

Case 3: ($y>1$)

$\Rightarrow 2^y-(2^{y-1}-1)=2^{y-1}+1\Rightarrow 2^y=2^{y-1}+2^{y-1}=2\left(2^{y-1}\right)=2^y$

So, all of $y>1,y\in$ Real Numbers

ie, This equation is satisfied $\forall y\in \left\{\right(-1\left)\right\}\cup [1,\infty )$