document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=206509>Question 206509</A>: <I><SPAN STYLE=\"word-break: break-all;\"> please show the conjecture is false by coming up with a counterexample.<BR>\n" );
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document.write( "the square root of a number x is always less than x. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #156048 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=156048\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%29%3Cx ALIGN=MIDDLE  ALT=\"sqrt%28x%29%3Cx\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given inequality.\r<BR>\n" );
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document.write( "To prove this false with a counter-example, we need to use a small number (that is less than 1). I'm going to use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F4 ALIGN=MIDDLE  ALT=\"1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F4%3C1 ALIGN=MIDDLE  ALT=\"1%2F4%3C1\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%281%2F4%29%3C1%2F4 ALIGN=MIDDLE  ALT=\"sqrt%281%2F4%29%3C1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1%2F4 ALIGN=MIDDLE  ALT=\"x=1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%281%29%2Fsqrt%284%29%3C1%2F4 ALIGN=MIDDLE  ALT=\"sqrt%281%29%2Fsqrt%284%29%3C1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the root.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fsqrt%284%29%3C1%2F4 ALIGN=MIDDLE  ALT=\"1%2Fsqrt%284%29%3C1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Evaluate the square root of 1 to get 1.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2%3C1%2F4 ALIGN=MIDDLE  ALT=\"1%2F2%3C1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Evaluate the square root of 4 to get 2.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2A4%3C1%2A2 ALIGN=MIDDLE  ALT=\"1%2A4%3C1%2A2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Cross multiply (this will help us determine which side is larger)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%3C2 ALIGN=MIDDLE  ALT=\"4%3C2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "Since the inequality is FALSE, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2%3C1%2F4 ALIGN=MIDDLE  ALT=\"1%2F2%3C1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > is FALSE (it turns out that one-half is actually larger than a quarter...draw out a picture to verify yourself). So this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%281%2F4%29%3C1%2F4 ALIGN=MIDDLE  ALT=\"sqrt%281%2F4%29%3C1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > is also false.\r<BR>\n" );
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document.write( "So we've shown that the inequality <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%29%3Cx ALIGN=MIDDLE  ALT=\"sqrt%28x%29%3Cx\"   DISABLED_style= \"max-width:90%; height:auto;\" > is false for all real numbers.\r<BR>\n" );
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document.write( "Note: it turns out that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%29%3Cx ALIGN=MIDDLE  ALT=\"sqrt%28x%29%3Cx\"   DISABLED_style= \"max-width:90%; height:auto;\" > is only true 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;\" > </SPAN>\n" );
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