If x<-2 then |1-|1+x|| equals

1974 AMC 12 Problem 15

Attempt Problem 15 of the 1974 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1974 AMC 12 solutions, or check the answer key.

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15.

x\lt-2

then

\bigl|1-|1+x|\bigr|

equals

2+x

-2-x

x

-x

-2

Resolve the inner absolute value first using

1+x\lt-1

After replacing

|1+x|

-1-x,

determine the sign of

2+x

Since

x\lt-2,

1+x\lt0,

|1+x|=-1-x.

Thus

\bigl|1-|1+x|\bigr|

becomes

|1-(-1-x)|,

which is

|2+x|=-2-x

because

2+x\lt0.

Therefore, the correct answer is B.