document.write( "Question 1208898: Prove that if 0 < a < b, then
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Algebra.Com's Answer #847408 by ikleyn(52794) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "Prove that if 0 < a < b, then
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document.write( "In the given inequality\r\n" );
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document.write( "    0 < a < b\r\n" );
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document.write( "divide all three terms by positive number ab.\r\n" );
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document.write( "    +-----------------------------------------------------------+\r\n" );
document.write( "    |      Use well known basic property of inequalities        |\r\n" );
document.write( "    |     that if all sides of the given/valid inequality       |\r\n" );
document.write( "    |   are multiplied or divided by the same positive number,  |\r\n" );
document.write( "    |      then the updated inequality is also valid. *)        |\r\n" );
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document.write( "You will get\r\n" );
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document.write( "    0/(ab) < a/((ab) < b/(ab).\r\n" );
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document.write( "Reduce common factors, and you will get the desired inequality\r\n" );
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document.write( "    0 < 1/b < 1/a.\r\n" );
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\n" ); document.write( "\n" ); document.write( "*)   This property of inequalities is a basic.\r
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\n" ); document.write( "\n" ); document.write( "        When you get an assignment like this one in the post,
\n" ); document.write( "        it is assumed that you just know the basic necessary prerequisites.\r
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\n" ); document.write( "\n" ); document.write( "comment from student:   You said: \". . .divide all three terms by positive number ab. \"
\n" ); document.write( "How did you know what to do here?
\n" ); document.write( "What information in the question indicates to divide all three by positive ab?\r
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\n" ); document.write( "\n" ); document.write( "My response:   You ask \" How did you know what to do here? \"\r
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\n" ); document.write( "\n" ); document.write( "For my solution and for your learning, it does not matter how I do know it.\r
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\n" ); document.write( "\n" ); document.write( "Probably, I learned it from my Math teacher/teachers; or from my school Math textbook/textbooks; or from many
\n" ); document.write( "other mathematical books that I read; or, may be, I was smart enouigh to invent this trick on my own.\r
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\n" ); document.write( "\n" ); document.write( "What $\"highlight%28highlight%28IS%29%29\"$ $\"highlight%28highlight%28really%29%29\"$ important - is the fact that it is $\"highlight%28highlight%28RIGHT%29%29\"$ way solving.\r
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\n" ); document.write( "\n" ); document.write( "        It is $\"highlight%28highlight%28RIGHT%29%29\"$ way solving.     It is $\"highlight%28highlight%28RIGHT%29%29\"$ way solving. \r
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\n" ); document.write( "\n" ); document.write( "Now, as you read it from my post, you also know this way thinking.\r
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\n" ); document.write( "\n" ); document.write( "It is how the process of spreading knowledge works: \r
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\n" ); document.write( "\n" ); document.write( "    - I learn something from some source;
\n" ); document.write( "    - I know that it is right - because it is my expert's suggestion;
\n" ); document.write( "    - I share and disseminate my knowledge;
\n" ); document.write( "    - after reading my post, you know it.\r
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\n" ); document.write( "\n" ); document.write( "So, your question \"how do I know it?\" is good, but it sounds in dissonance to my post.\r
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\n" ); document.write( "\n" ); document.write( "It is the same as after explanation in the class, given by your teacher,
\n" ); document.write( "you ask him or her: \"how do you know it?\" - In response, the teacher will look at you
\n" ); document.write( "and shrug his shoulders. - - - Simply because the teacher spent years of his life
\n" ); document.write( "to absorb knowledge and to learn the subject.\r
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\n" ); document.write( "\n" ); document.write( "In order for the communication be harmonic, you should say \"Thanks\" at the end.\r
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\n" ); document.write( "\n" ); document.write( "Then you will reach a harmony.\r
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\n" ); document.write( "\n" ); document.write( "In general, many of your comments remind me of an orchestra, where, when
\n" ); document.write( "the flutes are supposed to play, suddenly the sound of a drum is heard, instead.\r
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