document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=258748>Question 258748</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The difference between two positive numbers is 4&#8730;3. The product of the two<BR>\n" );
document.write( "numbers is 8. What is the absolute value of the difference of their reciprocals?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #190534 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=190534\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let a be the larger of the numbers.<BR>\n" );
document.write( "let b be the smaller of the numbers.\r<BR>\n" );
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document.write( "you have:\r<BR>\n" );
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document.write( "a-b = 4*sqrt(3)\r<BR>\n" );
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document.write( "you also have:\r<BR>\n" );
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document.write( "a*b = 8\r<BR>\n" );
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document.write( "the question is:\r<BR>\n" );
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document.write( " What is the absolute value of the difference of their reciprocals?\r<BR>\n" );
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document.write( "their reciprocals are:\r<BR>\n" );
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document.write( "(1/a) for a and (1/b) for b\r<BR>\n" );
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document.write( "since a > b, then 1/b > 1/a\r<BR>\n" );
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document.write( "this is derived as follows:\r<BR>\n" );
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document.write( "a > b\r<BR>\n" );
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document.write( "divide both sides of the equation by ab to get:\r<BR>\n" );
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document.write( "1/b > 1/a\r<BR>\n" );
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document.write( "an example would be 3 > 2.\r<BR>\n" );
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document.write( "divide both sides of this equation by 6 to get:\r<BR>\n" );
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document.write( "1/2 > 1/3\r<BR>\n" );
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document.write( "the absolute value of the difference of their reciprocals is:\r<BR>\n" );
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document.write( "|1/b - 1/a| \r<BR>\n" );
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document.write( "looking at the expression within the absolute value signs, we see:\r<BR>\n" );
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document.write( "(1/b - 1/a)\r<BR>\n" );
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document.write( "if we multiply this by (ab)/(ab), we get:\r<BR>\n" );
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document.write( "((a-b)/ab)\r<BR>\n" );
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document.write( "|1/b - 1/a| is therefore equal to |(a-b)/(ab)|\r<BR>\n" );
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document.write( "we know that a-b = 4*sqrt(3).\r<BR>\n" );
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document.write( "we also know that ab = 8\r<BR>\n" );
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document.write( "|1/b - 1/a| is therefore equal to |4*sqrt(3)/8|\r<BR>\n" );
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document.write( "this simplifies to |sqrt(3)/2|\r<BR>\n" );
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document.write( "since sqrt(3)/2 is always positive, then we have:\r<BR>\n" );
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document.write( "|1/b - 1/a| is equal to |sqrt(3)/2|\r<BR>\n" );
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document.write( "since sqrt(3)/2 is always positive, we have:\r<BR>\n" );
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document.write( "|1/b - 1/a| = |sqrt(3)/2| = sqrt(3)/2\r<BR>\n" );
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document.write( "this is because, by definition:\r<BR>\n" );
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document.write( "|x| = x if x is positive, and |x| = (-x) if x is negative.\r<BR>\n" );
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document.write( "your answer is:\r<BR>\n" );
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document.write( "the absolute value of the difference of their reciprocals equals sqrt(3)/2.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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