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April 15th, 2017, 01:20 PM   #1
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Irrational Inequation, how does one solve this?

x + (x2-10x+9)^(1/3) > [x+2(x2-10x+9)^(1/2)]^(1/2)

Solution: -45/4 < x < 1 or x >= 9

Last edited by skipjack; April 17th, 2017 at 02:54 AM.
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April 16th, 2017, 01:09 AM   #2
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Is there a typing error? The solution given is incorrect, as can be verified by substituting x = 0.
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April 16th, 2017, 09:58 AM   #3
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not sure, I don't even know what to do with it. It must be an error, but still, how do I solve the question then?
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April 16th, 2017, 03:04 PM   #4
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Where did you get the question from?

It appears that x + (x² - 10x + 9)^(1/2) > [x + 2(x² - 10x + 9)^(1/2)]^(1/2) was intended.
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April 16th, 2017, 04:21 PM   #5
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can no one solve this question ?
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April 16th, 2017, 05:33 PM   #6
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thank you guys!
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April 17th, 2017, 03:01 AM   #7
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Consider x + √(x² - 10x + 9) = √(x + 2√(x² - 10x + 9)).
Squaring both sides gives 2x² - 10x + 9 + 2x√(x² - 10x + 9) = x + 2√(x² - 10x + 9).
Rearranging that gives 2x² - 11x + 9 = 2√(x² - 10x + 9) - 2x√(x² - 10x + 9),
i.e., (x - 1)(2x - 9) = -2(x - 1)√(x² - 10x + 9).
Hence x = 1 or 2x - 9 = -2√(x² - 10x + 9).
Squaring that gives (2x - 9)² = 4(x² - 10x + 9), i.e. 4x = -45, so x = -45/4.

Solving the intended original inequation is now straightforward.
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