Problem 7 · 2013 Math Kangaroo
Medium
Algebra & Patterns
casework
We know that \(2 < x < 3\) for a number x. How many of the following four statements are then true?
\(4 < x^{2} < 9\) \(4 < 2x < 9\) \(6 < 3x < 9\) \(0 < x^{2} - 2x < 3\)
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Answer: E — 4
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Hint 1 of 2
Plug the range 2 < x < 3 into each inequality and check whether it must hold.
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Hint 2 of 2
For the last one, factor x² − 2x = x(x−2).
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Approach: bound each expression on (2,3)
- x in (2,3): x² in (4,9), so 4 < x² < 9 holds.
- 2x in (4,6) so 4 < 2x < 9 holds; 3x in (6,9) so 6 < 3x < 9 holds.
- x(x−2) with x in (2,3) gives a value in (0,3), so the fourth holds too — all four, answer E.
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