document.write( "Question 697295: If 4 < a < 7 < b < 9, then which of the following best defines a/b?\r
\n" ); document.write( "\n" ); document.write( "I tried to look at it like a triangle, with (a) being 6 and (b) being 8, but I'm stuck after that. \r
\n" ); document.write( "\n" ); document.write( "a. 4/9 < a/b < 1
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\n" ); document.write( "c. 4/7 < a/b < 7/9
\n" ); document.write( "d. 4/7 < a/b < 1
\n" ); document.write( "e. 4/7 < a/b < 9/7
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Algebra.Com's Answer #429870 by AnlytcPhil(1807)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Here is the correct solution:\r\n" );
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document.write( "First we must prove another theorem:\r\n" );
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document.write( "The reciprocals of two given unequal positive numbers are\r\n" );
document.write( "positive numbers unequal in the opposite order from the \r\n" );
document.write( "two given positive numbers.\r\n" );
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document.write( "[Maybe you've already proved this, but in case you haven't,\r\n" );
document.write( "here is the proof of that]:\r\n" );
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document.write( "Proof:\r\n" );
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document.write( "Suppose we are given:\r\n" );
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document.write( "0 < x < y  and we are to prove that 0 < \"1%2Fy\" < \"1%2Fx\"\r\n" );
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document.write( "Starting with\r\n" );
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document.write( "0 < x < y\r\n" );
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document.write( "divide all three sides by y:\r\n" );
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document.write( "\"0%2Fy\" < \"x%2Fy\" < \"y%2Fy\"\r\n" );
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document.write( "Simplify:\r\n" );
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document.write( "0 < \"x%2Fy\" < 1\r\n" );
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document.write( "Divide all three sides by x:\r\n" );
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document.write( "\"0%2Fx\" < \"expr%28x%2Fy%29\"÷\"x\" < \"1%2Fx\"\r\n" );
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document.write( "0 < \"expr%28x%2Fy%29\"÷\"x%2F1\" < \"1%2Fx\" \r\n" );
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document.write( "0 < \"x%2Fy\"·\"1%2Fx\" < \"1%2Fx\"\r\n" );
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document.write( "0 < \"cross%28x%29%2Fy\"·\"1%2Fcross%28x%29\" < \"1%2Fx\"\r\n" );
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document.write( "0 < \"1%2Fy\" < \"1%2Fx\"\r\n" );
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document.write( "Now that we have proved that, we can turn to your\r\n" );
document.write( "problem:\r\n" );
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document.write( "4 < a < 7 < b < 9\r\n" );
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document.write( "We are given \r\n" );
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document.write( "      4 < a  \r\n" );
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document.write( "Divide both sides by 9\r\n" );
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document.write( "(1)   \"4%2F9\" < \"a%2F9\" \r\n" );
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document.write( "We are given \r\n" );
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document.write( "      b < 9  \r\n" );
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document.write( "By the theorem above:\r\n" );
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document.write( "     \"1%2F9\" < \"1%2Fb\"\r\n" );
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document.write( "Multiply both sides by \"a\":\r\n" );
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document.write( "(2)   \"a%2F9\" < \"a%2Fb\"\r\n" );
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document.write( " \r\n" );
document.write( "We are given \r\n" );
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document.write( "      7 < b  \r\n" );
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document.write( "By the theorem above:\r\n" );
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document.write( "     \"1%2Fb\" < \"1%2F7\"\r\n" );
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document.write( "Multiply both sides by \"a\":\r\n" );
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document.write( "(3)   \"a%2Fb\" < \"a%2F7\"\r\n" );
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document.write( "We are given \r\n" );
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document.write( "      a < 7  \r\n" );
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document.write( "Divide both sides by 7\r\n" );
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document.write( "(4)   \"a%2F7\" < \"7%2F7\" \r\n" );
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document.write( "      \"a%2F7\" < 1\r\n" );
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document.write( "Putting (1), (2), (3) and (4) together:\r\n" );
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document.write( "      \"4%2F9\" < \"a%2Fb\" < \"a%2F7\" < 1\r\n" );
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document.write( "So omit \"a%2F7\" and we have\r\n" );
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document.write( "      \"4%2F9\" < \"a%2Fb\" < 1\r\n" );
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document.write( "as the only one of the inequalities given that \r\n" );
document.write( "we can be sure of.\r\n" );
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document.write( "Edwin
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