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Algebra Level 2

If 0 < a < 1 b < 1 , 0<a<\frac{1}{b}<1, then which of the following must be true?

a 2 > a > b > b 2 a^2 > a > b > b^2 b > a > a 2 > b 2 b > a > a^2 > b^2 b 2 > a > a 2 > b b^2 > a > a^2 >b b 2 > a 2 > b > a b^2 > a^2 > b > a b 2 > b > a > a 2 b^2 > b > a > a^2

Hana Wehbi by Hana Wehbi , 51, USA

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1 solution

Marta Reece
Aug 1, 2017

If 1 b < 1 \frac1b<1 and b > 0 b>0 then 1 < b 1<b

So we have positive a a and b b satisfying a < 1 < b a<1<b

Since a < 1 a<1 , a 2 = a × a < a × 1 = a a^2=a\times a<a\times1=a

And since b > 1 b>1 , b 2 = b × b > b × 1 = b b^2=b\times b>b\times1=b

Putting it together a 2 < a < b < b 2 \boxed{a^2<a<b<b^2}

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Thank you.

Hana Wehbi - 3 years, 10 months ago

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