document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=304976>Question 304976</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If x is a negative number, which of the following must be true? <BR>\n" );
document.write( "I) x^5 < |x| <BR>\n" );
document.write( "II) x < &#8730;(-x) <BR>\n" );
document.write( "III) x - 1/|x| < 0 \r<BR>\n" );
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document.write( "A) I, II and III <BR>\n" );
document.write( "B) I and II only <BR>\n" );
document.write( "C) None <BR>\n" );
document.write( "D) II and III only <BR>\n" );
document.write( "E) I and III only  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #218408 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=218408\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The best way to determine this is to use examples.\r<BR>\n" );
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document.write( "The first statement says that x^5 is smaller than the absolute value of x, shown as |x|.\r<BR>\n" );
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document.write( "This has to be true because x^5 is always a negative number and a negative number is always less than a positive number.\r<BR>\n" );
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document.write( "For example:\r<BR>\n" );
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document.write( "If x = -2, then x^5 = -32 which is smaller than 2.\r<BR>\n" );
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document.write( "If x = -1, then x^5 = -1 which is smaller than 1.\r<BR>\n" );
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document.write( "If x = -.2, then x^5 = -.00032 which is smaller than .2\r<BR>\n" );
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document.write( "You can't find a negative number that when raised to the 5th power is greater than or equal to a positive number, so this has to be a true statement for all negative numbers.\r<BR>\n" );
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document.write( "The second statement says that x is smaller than the square root of -x.\r<BR>\n" );
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document.write( "If x is negative, than -x is positive and the square root of -x is equal to plus or minus the square root of a positive number.\r<BR>\n" );
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document.write( "If you let x = -25, then:\r<BR>\n" );
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document.write( "Plus square root of 25 is equal to 5.   Since -25 is less than or equal to 5, this has to be true.\r<BR>\n" );
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document.write( "Minus square root of 25 is equal to -5.   Since -25 is less than or equal to -5, this also has to be true.\r<BR>\n" );
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document.write( "If you let x = -1, then:\r<BR>\n" );
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document.write( "Plus square root of 1 is equal to 1.   Since -1 is less than or equal to 1, this has to be true.\r<BR>\n" );
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document.write( "Minus square root of 1 is equal to -1.   Since -1 is NOT less than or equal to -01, this has to be false.\r<BR>\n" );
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document.write( "If you let x = -.25, then:\r<BR>\n" );
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document.write( "Plus square root of .25 is equal to .5.   Since -.25 is less than or equal to .5, this has to be true.\r<BR>\n" );
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document.write( "Minus square root of .25 is equal to -.5.   Since -.25 is NOT less than -.5, this has to be false.\r<BR>\n" );
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document.write( "Since you found at least one instance where a negative number is NOT less than the square root of minus the negative number, then this statement is false.\r<BR>\n" );
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document.write( "The third statement says that a negative number minus (1 divided by the absolute value of the negative number) is always less than 0.\r<BR>\n" );
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document.write( "If your number is -5, then 1/5 = .2 and -5 - .2 = -5.2.   Since -5.2 is less than 0, then this statement is true.\r<BR>\n" );
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document.write( "If your number is -1, then 1/1 = 1 and -1 - 1 = -2.   Since -2 is less than 0, then this statement is true.\r<BR>\n" );
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document.write( "If your number is -.5, then 1/.5 = 2 and -.5 - 2 = -2.5.   Since -2.5 is less than 0, then this statement is true.\r<BR>\n" );
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document.write( "Since you can't find a negative number where this statement is false, then this statement is always true.\r<BR>\n" );
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document.write( "Your answer appears to be selection E (I and III only).\r<BR>\n" );
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document.write( "You could not really generalize these statements to prove them true or not.   \r<BR>\n" );
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document.write( "You had to find examples and test them out.\r<BR>\n" );
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document.write( "The dividing line appears to be 1.\r<BR>\n" );
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document.write( "You had to test number whose absolute value of greater than 1, equal to 1, and less than 1.\r<BR>\n" );
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document.write( "Only testing with numbers greater than 1 would have led to a bad conclusion.\r<BR>\n" );
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document.write( "I almost did that with statement II until I tested with number whose absolute value was equal to 1 and less than 1.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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