document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1039635>Question 1039635</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve and shows the graph for the following problems.\r<BR>\n" );
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document.write( "-3 < 2/3x-9 <-1 ; x is not equal to 3 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #654396 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=654396\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i think you want to break this up into 2 parts.\r<BR>\n" );
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document.write( "first part is -3 < 2/(3x-9).\r<BR>\n" );
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document.write( "second part is 2/(3x-9) < -1.\r<BR>\n" );
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document.write( "there is a third piece that needs to be evaluated, which is where the vertical asymptote lies.\r<BR>\n" );
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document.write( "the vertical asymptote lies at x = 3.\r<BR>\n" );
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document.write( "when x < 3, the graph of 2/(3x-9) is negative and stays negative.\r<BR>\n" );
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document.write( "when x > 3, the graph of 2/(3x-9) is positive and stays positive.\r<BR>\n" );
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document.write( "this is clear from the equation, because when x > 3, (3x-9) is positive, and 2 is positive and therefore the fraction is positive.\r<BR>\n" );
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document.write( "this is also clear from the equation, because when x < 3, (3x-9) is negative, and therefore 2/(3x-9) is negative because the numerator is positive and the denominator is positive and therefore the fraction is negative.\r<BR>\n" );
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document.write( "what this tells you is that 2/(3x-9) is greater than -3 when x is greater than 3.\r<BR>\n" );
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document.write( "what this also tells you is that 2/(3x-9) is not smaller than -1 when x is greater than 3.\r<BR>\n" );
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document.write( "if you solve the equations for when -3 = 2/(3x-9) and when -1 = (2/(3x-9), you will find that:\r<BR>\n" );
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document.write( "-3 = 2/(3x-9) when x = 25/9 which is equal to 2.778 rounded to 3 decimal places.\r<BR>\n" );
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document.write( "-1 = 2/(3x-9) when x = 7/3 which is equal to 2.333 rounded to 3 decimal places.\r<BR>\n" );
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document.write( "at x = 25/9, 2/(3x-9) = -3<BR>\n" );
document.write( "when x < 25/9, 2/(3x-9) > -3.<BR>\n" );
document.write( "when x > 25/9, 2/(3x-9) < -3.<BR>\n" );
document.write( "however, when x > 3, 2/(3x-9) becomes positive and is therefore greater than -3 all over again.\r<BR>\n" );
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document.write( "therefore 2/(3x-9) is greater than -3 when x is smaller than 25/9 and when x is greater than 3.\r<BR>\n" );
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document.write( "this satisfies the inequality of -3 < 2/(3x-9).\r<BR>\n" );
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document.write( "the other inequality that needs to be satisfied is 2/(3x-9) < -1.\r<BR>\n" );
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document.write( "you know that 2/(3x-9) = -1 when x = 7/3.\r<BR>\n" );
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document.write( "if you look at the interval between x = 7/3 and x = 3, you will see that 3/(3x-9) is less than -1.\r<BR>\n" );
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document.write( "for example, when x = 8/3, which is greater than 7/3 and less than 3, 2/(3x-9 is equal to -30 which is clearly less than -1.\r<BR>\n" );
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document.write( "when x is less than 7/3, 2/(3x-9) is greater than -1.\r<BR>\n" );
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document.write( "for example, when x is equal to 6/3, which is less than 7/3, 2/(3x-9) is equal to -2/3 which is greater than -1.\r<BR>\n" );
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document.write( "therefore when x > 7/3 and < 3, y = 2/(3x-9) is less than -1.\r<BR>\n" );
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document.write( "what you really want to do is graph the equality points where the graph shows equality, and then look at the intervals between those points.\r<BR>\n" );
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document.write( "the following graph shows you what i mean.\r<BR>\n" );
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document.write( "i graphed 4 equations.\r<BR>\n" );
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document.write( "first equation is y = 2/(3x-9)<BR>\n" );
document.write( "second equation is y = -3<BR>\n" );
document.write( "third equation is y = -1<BR>\n" );
document.write( "fourth equation is x = 3.\r<BR>\n" );
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document.write( "you can see from the graph that y = -3 is smaller than y = 2/(3x-9) when x is smaller than 25/9 and when x is greater than 3.\r<BR>\n" );
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document.write( "you can also see from the graph that y = (3x-9) is smaller than y = -1x is greater than 7/3 and when x is smaller than 3.\r<BR>\n" );
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document.write( "those are your solutions.\r<BR>\n" );
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document.write( "-3 < 2/(3x-9) when x < 25/9 or when x > 3.<BR>\n" );
document.write( "in interval notation, this would be x = (minus infinity, 25/9) union (3, plus infinity).\r<BR>\n" );
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document.write( "2/(3x-9) < -1 when x > 7/3 and when x < 3.<BR>\n" );
document.write( "in interval notation, this would be (7/3,3).\r<BR>\n" );
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document.write( "here's the graph.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2016/062902.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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