document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=693458>Question 693458</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Select the counterexample that makes this conjecture false: \r<BR>\n" );
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document.write( "For any real number x,  x2 &#8805; x.\r<BR>\n" );
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document.write( "x = |3|<BR>\n" );
document.write( "x = -2<BR>\n" );
document.write( "x = 0<BR>\n" );
document.write( "x = 1/2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #427410 by </B><A HREF=/tutors/aboutme.mpl?userid=RedemptiveMath><B>RedemptiveMath(80)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=RedemptiveMath><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=427410\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A counterexample is an example that disproves a conjecture. It goes against what the conjecture states, and only one counterexample is needed to disprove a conjecture. The conjecture we are given is x^2 &#8805 x. It wants to know out of the list of possible counterexamples, which one is actually a counterexample? Again, for one of these values of x to be a counterexample it must make the statement x^2 &#8805 x false when we plug it in. So, we'll begin by plugging these values in for x:\r<BR>\n" );
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document.write( "|3|^2 &#8805 |3|<BR>\n" );
document.write( "3^2 &#8805 3<BR>\n" );
document.write( "9 &#8805 3 (9 is greater than 3, so this is not a counterexample)\r<BR>\n" );
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document.write( "(-2)^2 &#8805 (-2)<BR>\n" );
document.write( "4 &#8805 -2 (4 is greater than -2, so this is not a counterexample)\r<BR>\n" );
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document.write( "0^2 &#8805 0<BR>\n" );
document.write( "0 &#8805 0 (0 is equal to 0, so this is not a counter example)\r<BR>\n" );
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document.write( "(1/2)^2 &#8805 (1/2)<BR>\n" );
document.write( "1/4 &#8805 1/2\r<BR>\n" );
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document.write( "The last example shows that when x = 1/2, the conjecture x^2 &#8805 x is false because 1/4 is not greater than or equal to 1/2. This is what we would call a counterexample, and since we have found at least one counterexample, the conjecture x^2 &#8805 x is not a sound conjecture. That is, it is not always true. \r<BR>\n" );
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