document.write( "Question 213144: how to prove that (a+b)(b+c)(c+a)>8abc\r
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Algebra.Com's Answer #161133 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!
Prove that:
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document.write( "I think your problem must have said \r\n" );
document.write( "\"greater than or equal\", not \"greater than\"\r\n" );
document.write( "and also that a, b, and c \r\n" );
document.write( "are all non-negative.\r\n" );
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document.write( "\"Greater than\" is not true when a=b=c,\r\n" );
document.write( "because equality holds in that case\r\n" );
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document.write( "Substitute \"a\" for both \"b\" and \"c\":\r\n" );
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document.write( "\"%28a%2Ba%29%28a%2Ba%29%28a%2Ba%29=8aaa\"\r\n" );
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document.write( "That's true because it's equivalent to:\r\n" );
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document.write( "\"%282a%29%282a%29%282a%29=8a%5E3\"\r\n" );
document.write( "\"%282a%29%5E3=8a%5E3\"\r\n" );
document.write( "\"2%5E3%2Aa%5E3=8a%5E3\"\r\n" );
document.write( "\"8a%5E3=8a%5E3\"\r\n" );
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document.write( "It's also not necessarily true when 2 of them are\r\n" );
document.write( "negative.  So I think your problem must have said \r\n" );
document.write( "\"greater than or equal\", and also that a, b, and c \r\n" );
document.write( "are all non-negative.\r\n" );
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document.write( "First we prove the lemma: \r\n" );
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document.write( "\"a%2Bb+%3E=+2sqrt%28ab%29\"\r\n" );
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document.write( "Proof:\r\n" );
document.write( "\"%28a-b%29%5E2%3E=0\"       <----because the square of a real number is \r\n" );
document.write( "                            never negative\r\n" );
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document.write( "\"a%5E2-2ab%2Bb%5E2%3E=0\"   <----squaring out the left side\r\n" );
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document.write( "\"a%5E2%2B2ab%2Bb%5E2%3E=4ab\" <----adding \"4ab\" to both sides\r\n" );
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document.write( "\"%28a%2Bb%29%5E2%3E=4ab\"     <----factoring the left side \r\n" );
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document.write( "\"%28a%2Bb%29%3E=2sqrt%28ab%29\" <----taking non-negative square roots\r\n" );
document.write( "                            of both sides\r\n" );
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document.write( "Similarly we can prove \r\n" );
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document.write( "\"%28b%2Bc%29%3E=2sqrt%28bc%29\"\r\n" );
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document.write( "and\r\n" );
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document.write( "\"%28c%2Ba%29%3E=2sqrt%28ca%29\"\r\n" );
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document.write( "just by changing the letters in the first proof.\r\n" );
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document.write( "Therefore, multiplying the left and right sides of the\r\n" );
document.write( "three inequalities:\r\n" );
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document.write( "Edwin
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