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2001 AMC 10 Problem 2

2001 AMC 10 Problem 2AMC 10 Step by Step

Name: Date: September 13, 2026

Linear Equations & Word Problems · difficulty 1 of 5 · about 1 min · position P1-10

A number $x$ is $2$ more than the product of its reciprocal and its additive inverse. In which interval does the number lie?

A)
$-4\le x\le -2$
B)
$-2 < x\le 0$
C)
0< x \le 2
D)
$2 < x\le 4$
E)
$4 < x\le 6$

Answer

Key insight

The reciprocal 1/x times the additive inverse -x is always -1, so the number is simply -1 + 2 = 1.

Solution

The reciprocal of $x$ is $\frac{1}{x}$ and its additive inverse is $-x$ . Their product is
$\frac{1}{x}\cdot(-x) = -1,$
no matter what nonzero $x$ is. The sentence says $x$ is $2$ more than this product:
$x = -1 + 2 = 1.$
Since $0 < 1 \le 2$ , the number lies in interval (C).

The answer is $\boxed{\textbf{(C)}\ 0 < x \le 2}$ .

Why this works

Word problems that describe a number through its reciprocal and inverse often collapse because $x \cdot \frac1x = 1$ ; the "unknown" disappears from the product. Translate each phrase literally, simplify, and only then worry about the interval choices.

The trap

Translating 'is 2 more than' in the wrong direction (x + 2 = -1, giving x = -3 and choice A) or confusing reciprocal with additive inverse.

Common mistakes

Translating 'is 2 more than' in the wrong direction (x + 2 = -1, giving x = -3 and choice A) or confusing reciprocal with additive inverse.
Computing the product as $+1$ (forgetting the sign of the additive inverse) and getting $x = 3$ , choice (D).

Techniques

Set up the equation/formula and compute; no special trick needed