document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=308661>Question 308661</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Which of the following statements must be true when a^2 < b^2 and a and b are not 0?\r<BR>\n" );
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document.write( "I.    a^2/a < b^2/a<BR>\n" );
document.write( "II.  1/a^2 > 1/b^2<BR>\n" );
document.write( "III. (a + b) (a - b) < 0<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 #220928 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=220928\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I. Clearly this is false. If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3C0 ALIGN=MIDDLE  ALT=\"a%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then the inequality sign should flip, but it does not. If the sign did flip, then it would only be true for negative values of 'a', but what if 'a' was positive? Since we have this uncertainty, statement I is false.\r<BR>\n" );
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document.write( "II. If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3C%3E0 ALIGN=MIDDLE  ALT=\"a%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%3C%3E0 ALIGN=MIDDLE  ALT=\"b%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2 ALIGN=MIDDLE  ALT=\"a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2 ALIGN=MIDDLE  ALT=\"b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > are both positive numbers. Recall that if <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;\" > and 'x' and 'y' are both positive, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fx%3E1%2Fy ALIGN=MIDDLE  ALT=\"1%2Fx%3E1%2Fy\"   DISABLED_style= \"max-width:90%; height:auto;\" > which shows us that statement II is true.\r<BR>\n" );
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document.write( "III.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2+%3C+b%5E2 ALIGN=MIDDLE  ALT=\"a%5E2+%3C+b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given inequality.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2+-+b%5E2+%3C+0 ALIGN=MIDDLE  ALT=\"a%5E2+-+b%5E2+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2 ALIGN=MIDDLE  ALT=\"b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bb%29%28a-b%29+%3C+0 ALIGN=MIDDLE  ALT=\"%28a%2Bb%29%28a-b%29+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor the left side using the difference of squares.\r<BR>\n" );
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document.write( "So statement III is true.<BR>\n" );
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