Question

If $x<(-2)$, then$\left|1-\left|1+x\right|\right|$ equals to:

• A. $2+x$
• B. $x$
• C. $-x$
• D. $-(2+x)$

Answer 1

The correct option is D

$-(2+x)$

Solution:

Step 1: Find $\left|1+x\right|\mathrm{and}\left|2+x\right|$ for $x<-2$:

$$\begin{gathered} x<-2 \\ \Rightarrow x+1<-2+1(\mathrm{adding}1\mathrm{on}\mathrm{both}\mathrm{sides}) \\ \Rightarrow x+1<-1 \end{gathered}$$

$\Rightarrow (1+x)\in (-\infty ,-1)$

$\Rightarrow \left|1+x\right|=-(1+x)........\left(1\right)(\because \left|x\right|=-x,ifxisanegativenumber)$

Also,

$$\begin{gathered} x<-2 \\ \Rightarrow 2+x<-2+2(\mathrm{adding}2\mathrm{on}\mathrm{both}\mathrm{sides}) \\ \Rightarrow 2+x<0 \end{gathered}$$

$\Rightarrow (2+x)\in (-\infty ,0)$

$\Rightarrow \left|2+x\right|=-(2+x)........\left(2\right)(\because \left|x\right|=-x,ifxisanegativenumber)$

Step 2: Find $\left|1-\left|1+x\right|\right|$ for $x<-2$:

$$\begin{gathered} \left|1-\left|1+x\right|\right| \\ =\left|1-(-(1+x\left)\right)\right|\left(\mathrm{Using}\left(1\right)\right) \\ =\left|1+1+x\right| \\ =\left|2+x\right| \\ \mathbf{=}\mathbf{-}\mathbf{(}\mathbf{2}\mathbf{+}\mathit{x}\mathbf{)}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\left(\mathrm{Using}\left(2\right)\right) \end{gathered}$$

Final answer: Hence, the correct answer is option D.