1976 AMC 12 第 7 题

Question

若

x

为实数，则

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

为正数的充要条件是

If

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

或

-1\lt x\lt1

x\lt-1

or

-1\lt x\lt1

乘积为正数时，两个因子的符号相同

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

分别考察由

-1

和

1

分出的各个区间

Analyze the intervals cut by

-1

and

1

两个因子同时为正恰好发生在

|x|\lt1

时，此时

-1\lt x\lt1

。两个因子同时为负需要

|x|\gt1

且

x\lt-1

，这化为

x\lt-1

。两个端点都使乘积为零。

所以正确答案是 E。

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.

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