document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=348246>Question 348246</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let x,y belongs to Z. prove that if x>0 and x+y < xy,then y>0. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #249628 by </B><A HREF=/tutors/aboutme.mpl?userid=nyc_function><B>nyc_function(2741)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=nyc_function><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=nyc_function><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=249628\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Starting with the given,\r<BR>\n" );
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document.write( "x+y < xy\r<BR>\n" );
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document.write( "Subtract y from each side,\r<BR>\n" );
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document.write( "x < xy - y\r<BR>\n" );
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document.write( "x < y ( x - 1 )\r<BR>\n" );
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document.write( "Since x > 0, \r<BR>\n" );
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document.write( "0 < x < y(x-1)\r<BR>\n" );
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document.write( "0 < y(x-1)\r<BR>\n" );
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document.write( "By the trichotomy principle, y is either 0, negative, or positive.\r<BR>\n" );
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document.write( "Consider each case:\r<BR>\n" );
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document.write( "(a) y=0\r<BR>\n" );
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document.write( "0 < 0(x-1)=0 which is a contradiction.\r<BR>\n" );
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document.write( "(b) y<0\r<BR>\n" );
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document.write( "0 < y(x-1), divide by y to get\r<BR>\n" );
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document.write( "0 > (x-1) (recall that the inequality sign must change since we assumed y < 0)\r<BR>\n" );
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document.write( "1 > x, which is impossible since x is in Z and x > 0.\r<BR>\n" );
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document.write( "(c) The only possible choice, therefore, is y > 0.\r<BR>\n" );
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document.write( "That proves it.<BR>\n" );
document.write( " </SPAN>\n" );
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