document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199346>Question 1199346</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If a < b, show that a < (a + b)/2 < b. \r<BR>\n" );
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document.write( "Note: The number (a + b)/2 is called the arithmetic mean of a and b.\r<BR>\n" );
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document.write( "Can someone get me started by providing the first-two steps? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #833191 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=833191\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "a < (a+b)/2 < b<BR>\n" );
document.write( "breaks into<BR>\n" );
document.write( "a < (a+b)/2 and (a+b)/2 < b\r<BR>\n" );
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document.write( "Let's do a bit of algebraic operations on the first inequality we've broken down<BR>\n" );
document.write( "a < (a+b)/2<BR>\n" );
document.write( "a < 0.5(a+b)<BR>\n" );
document.write( "a < 0.5a+0.5b<BR>\n" );
document.write( "a-0.5a < 0.5b<BR>\n" );
document.write( "0.5a < 0.5b<BR>\n" );
document.write( "a < b\r<BR>\n" );
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document.write( "Reverse those steps:<BR>\n" );
document.write( "a < b<BR>\n" );
document.write( "0.5a < 0.5b<BR>\n" );
document.write( "0.5a+0.5a < 0.5b+0.5a<BR>\n" );
document.write( "a < 0.5a+0.5b<BR>\n" );
document.write( "a < 0.5(a+b)<BR>\n" );
document.write( "a < (a+b)/2\r<BR>\n" );
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document.write( "We can also say the following:<BR>\n" );
document.write( "a < b<BR>\n" );
document.write( "0.5a < 0.5b<BR>\n" );
document.write( "0.5a+0.5b < 0.5b+0.5b<BR>\n" );
document.write( "0.5(a+b) < b<BR>\n" );
document.write( "(a+b)/2 < b\r<BR>\n" );
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document.write( "With a < b as our starting point for the last two subsections, we've found that<BR>\n" );
document.write( "a < (a+b)/2 and (a+b)/2 < b<BR>\n" );
document.write( "which combines into<BR>\n" );
document.write( "a < (a+b)/2 < b\r<BR>\n" );
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document.write( "Therefore, the claim has been proven.\r<BR>\n" );
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document.write( "Another approach:\r<BR>\n" );
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document.write( "Let<BR>\n" );
document.write( "c = a+b\r<BR>\n" );
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document.write( "Add 'a' to both sides of the original inequality.<BR>\n" );
document.write( "a < b<BR>\n" );
document.write( "a+a < b+a<BR>\n" );
document.write( "2a < a+b<BR>\n" );
document.write( "2a < c\r<BR>\n" );
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document.write( "Now add 'b' to both sides<BR>\n" );
document.write( "a < b<BR>\n" );
document.write( "a+b < b+b<BR>\n" );
document.write( "c < 2b\r<BR>\n" );
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document.write( "Since 2a < c and c < 2b, we know that 2a < c < 2b.\r<BR>\n" );
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document.write( "Then as a last set of steps:<BR>\n" );
document.write( "2a < c < 2b<BR>\n" );
document.write( "2a < a+b < 2b<BR>\n" );
document.write( "2a/2 < (a+b)/2 < 2b/2<BR>\n" );
document.write( "a < (a+b)/2 < b<BR>\n" );
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