document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=296658>Question 296658</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If  x is an integer  greater  than  1  and  if  y  =  x + (1/x),  which of the following  must be true ?\r<BR>\n" );
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document.write( "I      y &#8800; x<BR>\n" );
document.write( "II     y is an integer <BR>\n" );
document.write( "III    xy > x^2\r<BR>\n" );
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document.write( "(A)  I  only<BR>\n" );
document.write( "(B)  III only<BR>\n" );
document.write( "(C)  I  and  II  only<BR>\n" );
document.write( "(D)  I  and III only<BR>\n" );
document.write( "(E)  I , II ,and III only<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 #213735 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=213735\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First, we need to see if statement I is true or not. Assume that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x ALIGN=MIDDLE  ALT=\"y=x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, if this is the case, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B1%2Fx=x ALIGN=MIDDLE  ALT=\"x%2B1%2Fx=x\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Subtract 'x' from both sides to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fx=0 ALIGN=MIDDLE  ALT=\"1%2Fx=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Since this equation is NEVER true for any 'x' values (let alone positive 'x' values), we can say that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x ALIGN=MIDDLE  ALT=\"y=x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is false where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%2B1%2Fx ALIGN=MIDDLE  ALT=\"y=x%2B1%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3C%3Ex ALIGN=MIDDLE  ALT=\"y%3C%3Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > and that statement I is true.\r<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E1 ALIGN=MIDDLE  ALT=\"x%3E1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fx ALIGN=MIDDLE  ALT=\"1%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT an integer. So this consequently means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B1%2Fx ALIGN=MIDDLE  ALT=\"x%2B1%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" > is also NOT an integer. So statement II is false meaning that choices C and E can be ignored.\r<BR>\n" );
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document.write( "Finally, because 'x' is an integer and 'x' is positive, we also know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3Cx%2B1%2Fx ALIGN=MIDDLE  ALT=\"x%3Cx%2B1%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" > or that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3Cy ALIGN=MIDDLE  ALT=\"x%3Cy\"   DISABLED_style= \"max-width:90%; height:auto;\" > (do a simple substitution here). Multiply both sides by 'x' to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%3Cxy ALIGN=MIDDLE  ALT=\"x%5E2%3Cxy\"   DISABLED_style= \"max-width:90%; height:auto;\" > (note: the sign doesn't flip because 'x' is positive).\r<BR>\n" );
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document.write( "So we've shown that if 'x' is an integer and 'x' is positive, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%3Cxy ALIGN=MIDDLE  ALT=\"x%5E2%3Cxy\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%2B1%2Fx ALIGN=MIDDLE  ALT=\"y=x%2B1%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So statement III is true.\r<BR>\n" );
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document.write( "So because statements I and III are both true, we can say that the answer is choice D.  </SPAN>\n" );
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