document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=711123>Question 711123</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If a, b, c are positive real numbers prove that<BR>\n" );
document.write( "(a + b + c)(1/a + 1/b + 1/c) >= 9\r<BR>\n" );
document.write( "\n" );
document.write( "I'm lost as to how to begin this proof </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #437332 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=437332\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First you multiply and simplify.<BR>\n" );
document.write( "I would also rearrange as shown below.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a+%2B+b+%2B+c%29%281%2Fa+%2B+1%2Fb+%2B+1%2Fc%29=a%281%2Fa%29%2Ba%2Fb%2Ba%2Fc%2Bb%2Fa%2Bb%281%2Fb%29%2Bb%2Fc%2Bc%2Fa%2Bc%2Fb%2Bc%281%2Fc%29=1%2Ba%2Fb%2Ba%2Fc%2Bb%2Fa%2B1%2Bb%2Fc%2Bc%2Fa%2Bc%2Fb%2B1=3%2B%28a%2Fb%2Bb%2Fa%29%2B%28a%2Fc%2Bc%2Fa%29%2B%28b%2Fc%2Bc%2Fb%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Next we have to show that each expression in brackets is equal or greater than 2.<BR>\n" );
document.write( "All of them are of the form<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Fy%2By%2Fx ALIGN=MIDDLE  ALT=\"x%2Fy%2By%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We could probe it with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Fy%2By%2Fx ALIGN=MIDDLE  ALT=\"x%2Fy%2By%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" > or with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Fb%2Bb%2Fa ALIGN=MIDDLE  ALT=\"a%2Fb%2Bb%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Fy%2By%2Fx=%28x%5E2%2By%5E2%29%2Fxy ALIGN=MIDDLE  ALT=\"x%2Fy%2By%2Fx=%28x%5E2%2By%5E2%29%2Fxy\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We want to prove that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2By%5E2%29%2Fxy%3E=2 ALIGN=MIDDLE  ALT=\"%28x%5E2%2By%5E2%29%2Fxy%3E=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2By%5E2%29-2xy=%28x-y%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%5E2%2By%5E2%29-2xy=%28x-y%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2By%5E2%29-2xy%3E=0 ALIGN=MIDDLE  ALT=\"%28x%5E2%2By%5E2%29-2xy%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2By%5E2%29%3E=2xy ALIGN=MIDDLE  ALT=\"%28x%5E2%2By%5E2%29%3E=2xy\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And since x and y are positive xy is a positive number we can use to divide both sides of the inequality to find that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2By%5E2%29%2Fxy%3E=2xy%2Fxy ALIGN=MIDDLE  ALT=\"%28x%5E2%2By%5E2%29%2Fxy%3E=2xy%2Fxy\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2By%5E2%29%2Fxy%3E=2 ALIGN=MIDDLE  ALT=\"%28x%5E2%2By%5E2%29%2Fxy%3E=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Fy%2By%2Fx%3E=2 ALIGN=MIDDLE  ALT=\"x%2Fy%2By%2Fx%3E=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Fb%2Bb%2Fa%3E=2 ALIGN=MIDDLE  ALT=\"a%2Fb%2Bb%2Fa%3E=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Fc%2Bc%2Fa%3E=2 ALIGN=MIDDLE  ALT=\"a%2Fc%2Bc%2Fa%3E=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2Fc%2Bc%2Fb%3E=2 ALIGN=MIDDLE  ALT=\"b%2Fc%2Bc%2Fb%3E=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a+%2B+b+%2B+c%29%281%2Fa+%2B+1%2Fb+%2B+1%2Fc%29=3%2B%28a%2Fb%2Bb%2Fa%29%2B%28a%2Fc%2Bc%2Fa%29%2B%28b%2Fc%2Bc%2Fb%29%3E=3%2B2%2B2%2B2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a+%2B+b+%2B+c%29%281%2Fa+%2B+1%2Fb+%2B+1%2Fc%29%3E=9 ALIGN=MIDDLE  ALT=\"%28a+%2B+b+%2B+c%29%281%2Fa+%2B+1%2Fb+%2B+1%2Fc%29%3E=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );