document.write( "Question 1207708: For 0 < a < b, let h be defined by (1/h) = (1/2)[(1/a) + (1/b)].
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Algebra.Com's Answer #845744 by mccravyedwin(408)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "I am tutor Edwin. I have two screen names (Edwin McCravy) and (mccravyedwin)\r\n" );
document.write( "Here is how I came up with my proof above. \r\n" );
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document.write( "Here is what I did on scratch paper, not something I would post,\r\n" );
document.write( "or hand in if I were a student.\r\n" );
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document.write( "I simplified this, getting it down to a \"flippable\" fraction:\r\n" );
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document.write( "\"1%2Fh+=+%281%2F2%29%28%281%2Fa%29+%2B+%281%2Fb%29%5E%22%22%29\"\r\n" );
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document.write( "\"1%2Fh+=+1%2F%282a%29+%2B+1%2F%282b%29\"\r\n" );
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document.write( "\"1%2Fh+=+%28b%2Ba%29%2F2ab\"\r\n" );
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document.write( "I flipped both sides:\r\n" );
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document.write( "\"h+=+%282ab%29%2F%28b%2Ba%29\"\r\n" );
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document.write( "Then (still on scratch paper), I started with the answer (which\r\n" );
document.write( "is 'illegal', but you can keep your scratch paper a secret.  Your\r\n" );
document.write( "teacher does not have to see it.  Anyway, I worked backwards from\r\n" );
document.write( "the answer, even though I had no right to do so. LOL\r\n" );
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document.write( "\"a\"\"%22%22%3C%22%22\"\"h\"\"%22%22%3C%22%22\"\"b\"\r\n" );
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document.write( "\"a\"\"%22%22%3C%22%22\"\"%282ab%29%2F%28b%2Ba%29\"\"%22%22%3C%22%22\"\"b\"\r\n" );
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document.write( "Then I multiplied through by (a+b), reversed from (b+a)\r\n" );
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document.write( "\"a%28a%2Bb%29\"\"%22%22%3C%22%22\"\"2ab\"\"%22%22%3C%22%22\"\"b%28a%2Bb%29\"\r\n" );
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document.write( "\"a%5E2%2Bab%29\"\"%22%22%3C%22%22\"\"2ab\"\"%22%22%3C%22%22\"\"ab%2Bb%5E2%29\"\r\n" );
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document.write( "Then I subtracted ab from all three sides:\r\n" );
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document.write( "\"a%5E2%29\"\"%22%22%3C%22%22\"\"ab\"\"%22%22%3C%22%22\"\"b%5E2%29\"\r\n" );
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document.write( "Then I noticed I could make two inequalities out of this:\r\n" );
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document.write( "\"a%5E2%29\"\"%22%22%3C%22%22\"\"ab\" and \"ab\"\"%22%22%3C%22%22\"\"b%5E2%29\"\r\n" );
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document.write( "I divided the first through by a, and the second one through by b,\r\n" );
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document.write( "\"a%29\"\"%22%22%3C%22%22\"\"b\" and \"a\"\"%22%22%3C%22%22\"\"b%29\"\r\n" );
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document.write( "They're both the same. And they're something that was given!!! \r\n" );
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document.write( "Now I must remind you that I had no right to start with the answer\r\n" );
document.write( "and work backwards.  However, by doing so anyway, I was able to reverse\r\n" );
document.write( "the process and get a process that I did have a right to do. \r\n" );
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document.write( "Now if you will compare what I just did with what I did above, you will \r\n" );
document.write( "see how I came up with my proof.  That is a method that sometimes helps\r\n" );
document.write( "you to see how to come up with a proof.  On scratch paper, start with\r\n" );
document.write( "the answer, work backwards, then see if you can reverse the steps.  Then\r\n" );
document.write( "toss your scratch paper in the waste basket, and your teacher will never\r\n" );
document.write( "know. [He/she might suspect though. LOL]\r\n" );
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document.write( "Edwin
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