Question

Find the solution to the equation
$2^{|x+2|}-|2^{x+1}-1|=2^{x+1}+1$

Answer 1

The various mode depend on the sign of x + 2 and x + 1 which gives x = -2 , -1
We consider the three cases
(I) x$\ge$-2
(II) -2 < x < -1
(III) x $\ge$ -1
The equation can be re - written as below the under different cases
(I) $2^{-(x+2)}-[-(2^{-(x+2)}-1\left)\right]=2^{-(x+2)}+1$
or $2^{-x-2}=2^1\therefore$\., - x - 2 = 1 or x = -3
This value satisfy I also
(II) $2^{-(x+2)}-[-(2^{-(x+2)}-1\left)\right]=2^{-(x+2)}+1$
or $2^{-x-2}=2^1$

$\therefore$ x + 2 = 1 or x = - 1
It is does not satisfy I
(III) $2^{-(x+2)}-[-(2^{-(x+2)}-1\left)\right]=2^{-(x+2)}+1$
or $2^{-(x+2)}=2,2^{-(x+2)}=2^{-(x+2)}$
Above is true x which is satisfy III
$\therefore x\ge$ -1 also are solution
x = -3 , x = $\ge$ -1