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2003 AMC 10A Problem 6

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2003 AMC 10A Problem 6AMC 10 Step by Step

Name: Date: September 13, 2026

Absolute Value & Inequalities · difficulty 1 of 5 · about 1 min · position P1-10

Define $x \heartsuit y$ to be $|x-y|$ for all real numbers $x$ and $y$ . Which of the following statements is not true?

A)
x \heartsuit y = y \heartsuit x $for all$ x $and$ y
B)
2(x \heartsuit y) = (2x) \heartsuit (2y) $for all$ x $and$ y
C)
x \heartsuit 0 = x $for all$ x
D)
x \heartsuit x = 0 $for all$ x
E)
$x \heartsuit y > 0$

Answer

Key insight

x heart 0 is |x|, which equals x only for nonnegative x; a single negative test value like x = -3 breaks statement (C).

Solution

Translate each statement using $x \heartsuit y = |x - y|$ .

(A) $|x - y| = |y - x|$ : true, since a number and its negative have the same absolute value.
(B) $2|x - y| = |2x - 2y|$ : true, because $|2t| = 2|t|$ .
(C) $x \heartsuit 0 = |x|$ . Taking $x = -3$ gives $|-3| = 3 \neq -3$ . False.
(D) $|x - x| = 0$ : true.
(E) $|x - y| > 0$ when $x \neq y$ : true, since only $0$ has absolute value $0$ .

The statement that fails is $\boxed{\textbf{(C)}\ x \heartsuit 0 = x\ \text{for all}\ x}$ .

Why this works

A "for all" claim is refuted by one counterexample, and the natural place to look with absolute values is a negative input, where $|x| = -x$ . Rewriting a made-up operation in terms of familiar functions makes each statement a known property to confirm or deny.

The trap

Forgetting that |x| = -x for negative x and believing (C) is true, then hunting for a flaw in a correct statement.

Common mistakes

Forgetting that |x| = -x for negative x and believing (C) is true, then hunting for a flaw in a correct statement.
Misreading (E) and testing $x = y$ , where the value is $0$ ; the statement only claims positivity when $x \neq y$ .

Techniques

Use the answer choices (mod checks, size, form) to eliminate or select · Test small/specific values or special cases to find or verify the answer