How to solve $ 2^{\vert x + 1 \vert} - 2^x = \vert 2^x - 1\vert + 1 $ for $ x $ I am new to modulus and inequalities , I came across this problem:
$ 2^{\vert x + 1 \vert} - 2^x = \vert 2^x - 1\vert + 1 $ for $ x $
How to find $ x $ ?

If $x\geq 0$, then


*

*$|x+1| = x+1$

*$|2^x-1| = 2^x-1$


So the equation becomes
$$2^{x+1}-2^x=2^x-1+1\\
2^{x+1} - 2\cdot 2^x=0\\
2^{x+1}-2^{x+1}=0\\
0=0$$
so it is always true.

There are two more cases to do: $-1\leq x<0$ and $x<-1$.
