document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1118504>Question 1118504</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If 0 < x < 1  ,  0 < y < 1 prove that 0 < x + y - xy < 1 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #733881 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=733881\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "x and y are both positive and less than 1; that means the product xy is less than x and less than y.<br><BR>\n" );
document.write( "So the expression x + y - xy is always positive; i.e., the inequality <br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C+x+%2B+y+-+xy ALIGN=MIDDLE  ALT=\"0+%3C+x+%2B+y+-+xy\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "is always true.<br><BR>\n" );
document.write( "To prove the other inequality<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%2B+y+-+xy+%3C+1 ALIGN=MIDDLE  ALT=\"x+%2B+y+-+xy+%3C+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "rewrite the statement to be proved as<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%2B+y+-+xy+-+1+%3C+0 ALIGN=MIDDLE  ALT=\"x+%2B+y+-+xy+-+1+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Then<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%281-y%29-%281-y%29+%3C+0 ALIGN=MIDDLE  ALT=\"x%281-y%29-%281-y%29+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%281-y%29+%3C+0 ALIGN=MIDDLE  ALT=\"%28x-1%29%281-y%29+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Because x and y are both between 0 and 1, one of those factors is always negative and the other is always positive, so the product is always negative.<br><BR>\n" );
document.write( "So the second inequality is also always true, making the original compound inequality always true. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );