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Problem 2

AMC 8 2000 Problem 2

AMC8 2000 年第 2 题

表达式与运算★★☆☆☆

Which of these numbers is less than its reciprocal?

下面哪个数比它的倒数小?

A.-2
B.-1
C.0
D.1
E.2

正确答案 · Correct Answer

解析 · Solution

Solution 1

The number 0 has no reciprocal, and 1 and -1 are their own reciprocals. This leaves only 2 and -2. The reciprocal of 2 is 1/2, but 2 is not less than 1/2. The reciprocal of -2 is -1/2, and -2 is less than -1/2, so it is

A

The statement "a number is less than its reciprocal" can be translated as

\displaystyle x < \frac{1}{x}

. Multiplication by

x

can be done if you do it in three parts: x>0, x=0, and x<0. You have to be careful about the direction of the inequality, as you do not know the sign of

x

. If x>0, the sign of the inequality remains the same. Thus, we have

x^2 < 1

when x > 0. This leads to 0 < x < 1. If x=0, the inequality

\displaystyle x < \frac{1}{x}

is undefined. If x<0, the sign of the inequality must be switched. Thus, we have

x^2 > 1

when x < 0. This leads to x < -1. Putting the solutions together, we have x<-1 or 0 < x < 1, or in interval notation,

(-\infty, -1) \cup(0, 1)

. The only answer in that range is -2

Starting again with

\displaystyle x < \frac{1}{x}

, we avoid multiplication by

x

. Instead, move everything to the left, and find a common denominator:

\displaystyle x < \frac{1}{x}

\displaystyle x - \frac{1}{x} < 0

\displaystyle \frac{x^2 - 1}{x} < 0

\displaystyle \frac{(x+1)(x-1)}{x} < 0

Divide this expression at x=-1, x=0, and x=1, as those are the three points where the expression on the left will "change sign". If x<-1, all three of those terms will be negative, and the inequality is true. Therefore,

(-\infty, -1)

is part of our solution set. If -1 < x < 0, the (x+1) term will become positive, but the other two terms remain negative. Thus, there are no solutions in this region. If 0 < x < 1, then both (x+1) and

x

are positive, while (x-1) remains negative. Thus, the entire region (0, 1) is part of the solution set. If 1 < x, then all three terms are positive, and there are no solutions. At all three "boundary points", the function is either 0 or undefined. Therefore, the entire solution set is

(-\infty, -1) \cup (0, 1)

, and the only option in that region is x=-2, leading to

A

We can find out all of their reciprocals. Now we compare and see that the answer is

A

Look at each number. Notice that the number must be negative. The number cannot be -1, 0, 1,.... -2 is all that is left

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