# Equation II

#### sbhatnagar

##### Active member
Solve the equation

$$2^{|x+2|}-|2^{x+1}-1|=2^{x+1}+1$$

#### Sudharaka

##### Well-known member
MHB Math Helper
Solve the equation

$$2^{|x+2|}-|2^{x+1}-1|=2^{x+1}+1$$
Hi sbhatnagar,

$|2^{x+1}-1| = \begin{cases}2^{x+1}-1 & \mbox{if } x \geq -1 \\\\ -2^{x+1}+1 & \mbox{if } x <-1 \end{cases}$

$|x+2|=\begin{cases}x+2 & \mbox{if } x \geq -2 \\\\ -x-2 & \mbox{if } x <-2 \end{cases}$

Therefore when $$x\geq -1$$ considering the left hand side of the equation we can obtain the right hand side.

$2^{x+2}-2^{x+1}+1=2.2^{x+1}-2^{x+1}+1=2^{x+1}+1$

That is the equation satisfies for each $$x\geq -1$$.

When $$-2\leq x<-1$$ we have,

$2^{x+2}+2^{x+1}-1=2^{x+1}+1$

$\Rightarrow 2^{x+2}=2$

Therefore the equation does not have a solution when $$-2\leq x<-1$$.

When $$x<-2$$,

$2^{-x-2}+2^{x+1}-1=2^{x+1}+1$

$\Rightarrow 2^{-x-2}=2$

$\therefore x=-3$

So the final solution is, $$x=[-1,\infty)\cup\{-3\}$$

Kind Regards,
Sudharaka.