document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207707>Question 1207707</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If 0 < a < b, show that a < sqrt{a•b} < b. The number sqrt{a•b} is called the geometric 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 #845740 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=845740\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "a < sqrt(ab) < b is the ultimate goal we want to reach.\r<BR>\n" );
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document.write( "Break it into these two key parts<BR>\n" );
document.write( "a < sqrt(ab) and sqrt(ab) < b\r<BR>\n" );
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document.write( "Let's focus on the first part<BR>\n" );
document.write( "a < sqrt(ab)<BR>\n" );
document.write( "a^2 < ab ......... square both sides<BR>\n" );
document.write( "a < b ........... divide both sides by 'a'\r<BR>\n" );
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document.write( "The inequality sign won't flip when dividing both sides by 'a' since a > 0.\r<BR>\n" );
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document.write( "Let's reverse the flow of that logic to get this:<BR>\n" );
document.write( "a < b<BR>\n" );
document.write( "a*a < b*a<BR>\n" );
document.write( "a^2 < ab<BR>\n" );
document.write( "sqrt( a^2 ) < sqrt(ab)<BR>\n" );
document.write( "a < sqrt(ab)\r<BR>\n" );
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document.write( "Follow a similar set of steps for the other key part.<BR>\n" );
document.write( "a < b<BR>\n" );
document.write( "a*b < b*b<BR>\n" );
document.write( "ab < b^2<BR>\n" );
document.write( "sqrt(ab) < sqrt(b^2)<BR>\n" );
document.write( "sqrt(ab) < b\r<BR>\n" );
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document.write( "We have determined <BR>\n" );
document.write( "a < sqrt(ab) and sqrt(ab) < b<BR>\n" );
document.write( "which glue together to get<BR>\n" );
document.write( "a < sqrt(ab) < b<BR>\n" );
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