document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207706>Question 1207706</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If a < b, show that a < (a + b)/2 < b. The number (a + b)/2 is called the arithmetic mean of a and b.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #845741 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=845741\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "We want to show that a < (a + b)/2 < b\r<BR>\n" );
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document.write( "This breaks down into these two key parts<BR>\n" );
document.write( "a < (a + b)/2  and  (a + b)/2 < b\r<BR>\n" );
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document.write( "What I'll do is start with the first piece and follow these steps<BR>\n" );
document.write( "a < (a + b)/2<BR>\n" );
document.write( "2a < a + b<BR>\n" );
document.write( "2a-a < a + b - a<BR>\n" );
document.write( "a < b\r<BR>\n" );
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document.write( "Next, I'll reverse those steps by following the chain upward.<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( "a < (a+b)/2\r<BR>\n" );
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document.write( "We have gone from the given a < b to prove that a < (a+b)/2 is the case.\r<BR>\n" );
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document.write( "Follow similar steps for the other needed key component.<BR>\n" );
document.write( "a < b <BR>\n" );
document.write( "a+b < b+b <BR>\n" );
document.write( "a+b < 2b <BR>\n" );
document.write( "(a+b)/2 < b\r<BR>\n" );
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document.write( "We have shown that if a < b, then <BR>\n" );
document.write( "a < (a + b)/2  and  (a + b)/2 < b<BR>\n" );
document.write( "in which those two pieces glue together to get<BR>\n" );
document.write( "a < (a + b)/2 < b\r<BR>\n" );
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document.write( "--------------------------------------------------------------------------\r<BR>\n" );
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document.write( "Another approach\r<BR>\n" );
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document.write( "Let 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( "Return back to the original inequality.<BR>\n" );
document.write( "This time 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,<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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