Question

The solution of the equation $2^{|x+1|}-2^x=|2^x-1|+1$ is union of

• A. a constant and $x>0$
• B. a constant and $x\ge 0$
• C. a constant and $x\le 0$
• D. a constant and $x<0$

Answer 1

Answer: (B)

a constant and $x\ge 0$

We have,
$2^{|x+1|}-2^x=|2^x-1|+1$
$2^x-1>0$ , for $x>0$ and $2^x-1<0$ for $x<0$.
To solve this let us consider various cases.
Case 1 : $x\ge 0$
$2^{x+1}-2^x=2^x-1+1$
$2^{x+1}-2\times 2^x=0$
$0=0$
This is true.
Hence, $x\ge 0$ satisfies the given equation.
Case 2 : $-1\le x<0$
The equation can be simplified to :
$2^{x+1}-2^x=1-2^x+1$
$\Rightarrow 2^{x+1}=2$
$x=0$
This solution has already been included earlier.
Case 3 : $x<-1$
$2^{-1-x}-2^x=1-2^x+1$
$\Rightarrow 2^{-1-x}=2$
$-1-x=1$
$x=-2$
Hence, the solution set of $x$ is $[0,\infty )\cup \{-2\}$