Problem: If x<−2x<-2x<−2 then ∣1−∣1+x∣∣|1-|1+x||∣1−∣1+x∣∣ equals
Answer Choices:
A. 2+x2+x2+x
B. −2−x-2-x−2−x
C. xxx
D. −x-x−x
E. −2-2−2 Solution:
By definition
∣a∣={aa≥0−aa<0|a|=\left\{\begin{array}{rl}a & a \geq 0 -a & a<0\end{array}\right. ∣a∣={aa≥0−aa<0
If x<−2x<-2x<−2, then 1+x<01+x<01+x<0 and 11+x+=−(1+x)11+x+=-(1+x)11+x+=−(1+x) and ∣1−∣1+x∣∣=∣1+1+x∣=∣2+x∣|1-|1+x||=|1+1+x|=|2+x |∣1−∣1+x∣∣=∣1+1+x∣=∣2+x∣. Again if x<−2x<-2x<−2, then 2+x<02+x < 02+x<0 and ∣2+x∣=−2−x|2+x|=-2-x∣2+x∣=−2−x.