document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207708>Question 1207708</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For 0 < a < b, let h be defined by (1/h) = (1/2)[(1/a) + (1/b)].<BR>\n" );
document.write( "Show that a < h < b. The number h is called the harmonic mean of a and b.\r<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 #845738 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=845738\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "0 < a < b<BR>\n" );
document.write( "a < b<BR>\n" );
document.write( "1/a > 1/b <BR>\n" );
document.write( "1/b < 1/a<BR>\n" );
document.write( "1/b < 1/h < 1/a ..... see note below<BR>\n" );
document.write( "b > h > a <BR>\n" );
document.write( "a < h < b\r<BR>\n" );
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document.write( "Note:  1/h is the midpoint of 1/a and 1/b, due to the formula your teacher gave you.<BR>\n" );
document.write( "Therefore, 1/h is guaranteed to be between those values. <BR>\n" );
document.write( "A way to prove this can be found through <a  rel=nofollow HREF=\"https://www.algebra.com/algebra/homework/Inequalities/Inequalities.faq.question.1207706.html\">this question</a>\r<BR>\n" );
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document.write( "Don't forget to flip the inequality signs when applying reciprocals to all sides. \r<BR>\n" );
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document.write( "Tutor Edwin has a great approach. I'll paraphrase his method slightly.\r<BR>\n" );
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document.write( "Let's rewrite the given formula<BR>\n" );
document.write( "1/h = (1/2)*(1/a + 1/b)<BR>\n" );
document.write( "1/h = (1/2)*(b/(ab) + a/(ab))<BR>\n" );
document.write( "1/h = (1/2)*( (a+b)/(ab) )<BR>\n" );
document.write( "1/h = (a+b)/(2ab)<BR>\n" );
document.write( "h = 2ab/(a+b)<BR>\n" );
document.write( "I'll come back to this later.\r<BR>\n" );
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document.write( "Then,<BR>\n" );
document.write( "a < b turns into a^2 < ab after multiplying both sides by a<BR>\n" );
document.write( "a < b turns into ab < b^2 after multiplying both sides by b<BR>\n" );
document.write( "Both a and b are positive, so there's no inequality sign flips when multiplication is applied to both sides.<BR>\n" );
document.write( "a^2 < ab  &  ab < b^2 combine to a^2 < ab < b^2\r<BR>\n" );
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document.write( "and finally<BR>\n" );
document.write( "a^2 < ab < b^2<BR>\n" );
document.write( "a^2+ab < ab+ab < b^2+ab ...... adding ab to all sides<BR>\n" );
document.write( "a^2+ab < 2ab < b^2+ab<BR>\n" );
document.write( "a(a+b) < 2ab < b(a+b)<BR>\n" );
document.write( "a < 2ab/(a+b) < b ........... dividing all sides by (a+b)<BR>\n" );
document.write( "a < h < b\r<BR>\n" );
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document.write( "Since a > 0 and b > 0, we can be certain that a+b > 0 as well of course.<BR>\n" );
document.write( "a+b > 0 means dividing both sides by (a+b) will not flip the inequality sign.<BR>\n" );
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