AMC 10 · 2003 · #6

Grade 6 arithmetic
Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In
absolute-valueif-then-reasoning casework ↑ Prerequisites: absolute-value
📏 Medium solution 💡 2 insights
Problem
A new operation ♥ is defined by x ♥ y = |x-y| for all real numbers x and y. Four of the five statements (A)–(E) about this operation are always true; find the one statement that is not true.

Pick an answer.

(A)
$x \heartsuit y = y \heartsuit x \text{ for all } x, y$
(B)
$2(x \heartsuit y) = (2x) \heartsuit (2y) \text{ for all } x, y$
(C)
$x \heartsuit 0 = x \text{ for all } x$
(D)
$x \heartsuit x = 0 \text{ for all } x$
(E)
$x \heartsuit y > 0 \text{ if } x \neq y$
How to solve
Strategy Eliminate Possibilities

The question asks which statement fails, so tool #3 (Eliminate Possibilities) fits exactly: test each of the five claims, throw out the ones that always hold, and the survivor is the answer. Because x ♥ y = |x-y| is just the distance between x and y, four of the claims are quick distance facts (symmetry, scaling, zero-distance, positive distance) and confirm as true. Tool #6 (Guess and Check) drives the disproof: a statement claiming something 'for all x' is broken by a single number that fails it. Tool #14 (Extreme Principle) tells us where to look for that number — absolute value only changes things for negatives, so probing a negative input is the case most likely to expose a false claim, and statement (C) breaks there.

1STEP 1

Read the heart as distance

The operation is just the distance between two numbers, so it is never negative.

x ♥ y = |x-y| = distance from x to y ≥ 0
2STEP 2

Confirm A, B, D, E are true

Four of the claims survive: distance is symmetric, scales by two, is zero at equal inputs, and is positive otherwise — all true.

|x-y|=|y-x|; |2x-2y|=2|x-y|; |x-x|=0; x ≠ y → |x-y| ＞ 0
3STEP 3

Break statement C with a negative

But the fifth claims the operation with zero returns the number itself; at negative three it returns 3, not negative three.

x ♥ 0 = |x|; (-3)♥ 0 = |-3| = 3 ≠ -3 → (C)
Answer
x ♥ 0 = x for all x
The correct identity is x ♥ 0 = |x|, which equals x for non-negative x but equals -x for negative x. So statement (C) is true for half the number line and false for the other half — exactly the kind of 'almost right' claim that a 'for all' statement cannot afford. Every other statement is a genuine property of distance (symmetry, scaling, zero at equal points, positive when unequal), so (C) being the lone false one is consistent.
💡Key takeaway

A statement that says something is true 'for all' numbers is wrong the moment one number breaks it, and with absolute value the number to try is a negative one.

  • Read the heart as distance
  • Confirm A, B, D, E are true
  • Break statement C with a negative

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