document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=807631>Question 807631</A>: <I><SPAN STYLE=\"word-break: break-all;\"> the product of two consecutive odd integers is less than 5 times their sum<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 #486435 by </B><A HREF=/tutors/aboutme.mpl?userid=DrBeeee><B>DrBeeee(684)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=DrBeeee><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=486435\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The trick you need to learn about integer problems is \"How do we represent them.\"<BR>\n" );
document.write( "Let n = the first odd integer<BR>\n" );
document.write( "Then the NEXT consecutive ODD integer is (n + 2). Not n+1, because if n is odd then n+1 would be even! <BR>\n" );
document.write( "So<BR>\n" );
document.write( "Let n+2 = the next odd integer<BR>\n" );
document.write( "The problem statement is given by<BR>\n" );
document.write( "(1) n*(n+2) < 5*(n + n +2) or<BR>\n" );
document.write( "(2) n^2 + 2n < 10n + 10 or<BR>\n" );
document.write( "(3) n^2 + 2n - 10n < 10 or<BR>\n" );
document.write( "(4) n^2 - 8n < 10 or<BR>\n" );
document.write( "(5) n*(n-8) < 10<BR>\n" );
document.write( "The form of (5) is really arbitrary and not super important. <BR>\n" );
document.write( "What we need to do is find the values of n (remember it is an odd integer) that satisfy the inequality in (5). I use trial and error.<BR>\n" );
document.write( "First I start a -13, then we get<BR>\n" );
document.write( "(6) -13*(-21) = 273 which is not less than 10<BR>\n" );
document.write( "So let's move to a higher value like n = -3, where we get<BR>\n" );
document.write( "(7) -3*(-11) = 33, stiil not less than 10.<BR>\n" );
document.write( "Let's try n = -1, and get<BR>\n" );
document.write( "(8) -18(-9) = 9. Aha, the least integer is <BR>\n" );
document.write( "(9) n > -1<BR>\n" );
document.write( "Now go to high positive odd integers, say n = 11 and get<BR>\n" );
document.write( "(10) 11*(3) = 33 which is not less than 10. So try n = 9 and get<BR>\n" );
document.write( "(11) 9*(1) = 9. Aha we hit another limit.<BR>\n" );
document.write( "To be sure there's a range of n's, let's try n = 7 and get<BR>\n" );
document.write( "(12) 7*(-1) = -7 which is less than 10, so is OK. Let's check the low value at n = 1 and get<BR>\n" );
document.write( "(13) 1*(-7) = -7, so we have<BR>\n" );
document.write( "(14)<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-1%3C=n%3C=9 ALIGN=MIDDLE  ALT=\"+-1%3C=n%3C=9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Answer: The consecutive odd integers are (-1,1), (1,3), (3,5), (5,7), (7,9), (9,11).<BR>\n" );
document.write( "Note: we can solve for the roots of the equation <BR>\n" );
document.write( "(15) n^2 - 8n - 10 = 0 and get <BR>\n" );
document.write( "(16) n = 9.1 and -1.1<BR>\n" );
document.write( "Since we want intger values for n we choose<BR>\n" );
document.write( "(17) n = [-1,9], the same as my lenghty trial and error. <BR>\n" );
document.write( "PS I actually found the roots first.\r<BR>\n" );
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