Problem 12 · 2012 Math Kangaroo
Medium
Algebra & Patterns
casework
A real number x fulfills the condition \(x^3 < 64 < x^2\). Which of the following statements is definitely true?
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Answer: E — \(x < -8\)
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Hint 1 of 2
Split the chain into x³ < 64 and 64 < x² separately.
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Hint 2 of 2
One gives an upper bound on x; the other forces x to be very negative.
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Approach: solve each inequality and intersect
- \(x^3 < 64\) means \(x < 4\).
- \(64 < x^2\) means \(|x| > 8\), i.e. \(x > 8\) or \(x < -8\).
- The only overlap with \(x < 4\) is \(x < -8\), so choice E is definitely true.
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