1976 AMC 12 Problem 7

Attempt Problem 7 of the 1976 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 1976 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

x

is a real number, then the quantity

(1-|x|)(1+x)

is positive if and only if

|x|\lt1

x\lt1

|x|\gt1

x\lt-1

x\lt-1

-1\lt x\lt1

A product is positive when its two factors have the same sign

Analyze the intervals cut by

-1

and

1

Both factors are positive exactly when

|x|\lt1,

giving

-1\lt x\lt1.

Both are negative when

|x|\gt1

and

x\lt-1,

which reduces to

x\lt-1.

The endpoints make the product zero.

Therefore, the correct answer is E.

Problem 7 in other years