MHB Solving Equation: $2^{|x+2|}-|2^{x+1}-1|=2^{x+1}+1$

[sbhatnagar]

Solve the equation

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

 

[Sudharaka]

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.