Question

Solve the equation
$2^{|x+1|}-2^x=|2^x-1|+1$
$xϵ(a,\infty )\cup (-b)$
Find $a+b$

Answer 1

Find critical points:
$x+1=0,2^x-1=0\therefore x=-1,x=0$
consider following cases:
x<-1:
$2^{-(x+1)}-2^x=(2^x-1)+1\Rightarrow 2^{-(x+1)}=2\therefore -(x+1)=1\therefore x=-\mathrm{2............}\left(1\right)$
$-1\le x<0$:
$2^{x+1}-2^x=(2^x-1)+1\Rightarrow 2^{x+1}=2\therefore x+1=1\therefore x=0x\ne 0(\because -1\le x<0)$
$x\ge 0:$
$2^{x+1}-2^x=2^x-1+1\Rightarrow 2^{x+1}=2.2^x\Rightarrow 2^{x+1}=2^{x+1}$
which is true for $x\ge 0$
Now combining all cases, we have the final solution as:
$xϵ(0,\infty )\cup (-2)$
Ans: 2