Problem 20 · AMC 8 Stretch
Stretch
Algebra & Patterns
account-for-all-possibilitieslogical-reasoning
How many numbers \(x\) make \(|x - 2| + |x - 6| < 3\) true? (If none, answer \(0\).)
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Answer: 0 (no solution)
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Hint 1 of 4
Think about what \(|x - 2|\) and \(|x - 6|\) mean: the distance from \(x\) to \(2\), and the distance from \(x\) to \(6\). So the left side is (distance to \(2\)) + (distance to \(6\)).
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Hint 2 of 4
The points \(2\) and \(6\) are \(4\) apart. If \(x\) sits between them, the two distances must add up to exactly \(4\). Could that ever be less than \(3\)?
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Hint 3 of 4
If \(x\) is outside the interval (below \(2\) or above \(6\)), the two distances add up to MORE than \(4\). So the sum is never smaller than \(4\).
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Approach: Read absolute values as distances on a number line
- Read the left side as distances: \(|x - 2|\) is how far \(x\) is from \(2\), and \(|x - 6|\) is how far \(x\) is from \(6\). The left side is the total distance to both points.
- The points \(2\) and \(6\) are \(4\) apart. If \(x\) is between them, the two distances split the gap and add to exactly \(4\); if \(x\) is outside, the total is even more than \(4\).
- So no matter what \(x\) is, the left side is always at least \(4\), and can never be less than \(3\).
- Therefore there is no solution: \(0\) values of \(x\) work.
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