document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=98471>Question 98471</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi there,\r<BR>\n" );
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document.write( "I'm having difficulty solving this inequality:\r<BR>\n" );
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document.write( "3(t-1)(t-3) > 0\r<BR>\n" );
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document.write( "So at first I tried isolating it looks like:\r<BR>\n" );
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document.write( "3 > 0 , (t-1) > 0 , (t-3) > 0\r<BR>\n" );
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document.write( "In this sense, I would get:<BR>\n" );
document.write( "t > 1 or t > 3\r<BR>\n" );
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document.write( "But the right answer is supposed to be t > 3 (Ok, I get this one I think), and t < 1.  How did they get the last inequality? \r<BR>\n" );
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document.write( "Thank you very much! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #71653 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=71653\">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=3%28t-1%29%28t-3%29+%3E+0 ALIGN=MIDDLE  ALT=\"3%28t-1%29%28t-3%29+%3E+0\"   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=3%28t-1%29%28t-3%29+=+0 ALIGN=MIDDLE  ALT=\"3%28t-1%29%28t-3%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the left side equal to zero\r<BR>\n" );
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document.write( "Now set each factor equal to zero:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t-3=0 ALIGN=MIDDLE  ALT=\"t-3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t-1=0 ALIGN=MIDDLE  ALT=\"t-1=0\"   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=t=3 ALIGN=MIDDLE  ALT=\"t=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=1 ALIGN=MIDDLE  ALT=\"t=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Now solve for t in each case\r<BR>\n" );
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document.write( "So our solutions are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=3 ALIGN=MIDDLE  ALT=\"t=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=1 ALIGN=MIDDLE  ALT=\"t=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "These values make the entire inequality equal to zero. So that means that at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=3 ALIGN=MIDDLE  ALT=\"t=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=1 ALIGN=MIDDLE  ALT=\"t=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > the graph will go from positive to negative (or vice versa). So we must use these points to test the inequality\r<BR>\n" );
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document.write( "So now pick any value less than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=1 ALIGN=MIDDLE  ALT=\"t=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and test the inequality <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28t-1%29%28t-3%29+%3E+0 ALIGN=MIDDLE  ALT=\"3%28t-1%29%28t-3%29+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So lets pick t=0\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%280-1%29%280-3%29+%3E+0 ALIGN=MIDDLE  ALT=\"3%280-1%29%280-3%29+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in t=0\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28-1%29%28-3%29+%3E+0 ALIGN=MIDDLE  ALT=\"3%28-1%29%28-3%29+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%3E+0 ALIGN=MIDDLE  ALT=\"9%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "Since t=0 makes the inequality true, that means everything less than 1 will make the inequality true. So one part of our answer is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%3C1 ALIGN=MIDDLE  ALT=\"t%3C1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now pick any point in between  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=1 ALIGN=MIDDLE  ALT=\"t=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >  and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=3 ALIGN=MIDDLE  ALT=\"t=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Let's pick t=2\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%282-1%29%282-3%29+%3E+0 ALIGN=MIDDLE  ALT=\"3%282-1%29%282-3%29+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in t=2\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%281%29%28-1%29+%3E+0 ALIGN=MIDDLE  ALT=\"3%281%29%28-1%29+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3%3E+0 ALIGN=MIDDLE  ALT=\"-3%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "Since t=2 makes the inequality false, that means the possible answer <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%3E1 ALIGN=MIDDLE  ALT=\"t%3E1\"   DISABLED_style= \"max-width:90%; height:auto;\" > false. So this shows why <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%3E1 ALIGN=MIDDLE  ALT=\"t%3E1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is not an answer.\r<BR>\n" );
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document.write( "Now pick any point greater than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=3 ALIGN=MIDDLE  ALT=\"t=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Let's pick t=4\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%284-1%29%284-3%29+%3E+0 ALIGN=MIDDLE  ALT=\"3%284-1%29%284-3%29+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in t=0\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%283%29%281%29+%3E+0 ALIGN=MIDDLE  ALT=\"3%283%29%281%29+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%3E+0 ALIGN=MIDDLE  ALT=\"9%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "Since t=4 makes the inequality true, that means one part of our answer is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%3E3 ALIGN=MIDDLE  ALT=\"t%3E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "That last part is somewhat unnecessary since we already eliminated everything else. However, it is important to see everything that is going on\r<BR>\n" );
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document.write( "So our solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%3C1 ALIGN=MIDDLE  ALT=\"t%3C1\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%3E3 ALIGN=MIDDLE  ALT=\"t%3E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Notice if we graph the equation we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+3%28x-1%29%28x-3%29%29+ ALIGN=MIDDLE  ALT=\"+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+3%28x-1%29%28x-3%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28t-1%29%28t-3%29 ALIGN=MIDDLE  ALT=\"3%28t-1%29%28t-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and we can clearly see that everything to the left of 1 is greater than zero. Also, we can see that everything to the right of 3 is greater than zero. So our answer is verified. </SPAN>\n" );
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