document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=5387>Question 5387</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am sending two questions, because I think they are very similar, although I was able to solve only the first one, but in a somewhat unsatisfactory way:\r<BR>\n" );
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document.write( "1)  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%281%2Flog%282%2Cx%29%29%3C=%281%2Flog%282%2Csqrt%28x%2B2%29%29%29+ ALIGN=MIDDLE  ALT=\"+%281%2Flog%282%2Cx%29%29%3C=%281%2Flog%282%2Csqrt%28x%2B2%29%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "2)  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%281%2Flog%282%2Cx%29%29-%281%2F%28log%282%2Cx%29%29-1%29%3C1++ ALIGN=MIDDLE  ALT=\"+%281%2Flog%282%2Cx%29%29-%281%2F%28log%282%2Cx%29%29-1%29%3C1++\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "My goal would be to solve these inequalities purely by algebraic manipulation.\r<BR>\n" );
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document.write( "I examined both questions with Mathematica 4 and graphed them with Winplot (Peanut)and both confirm the answers, that by coincidence are  the same for the two questions.\r<BR>\n" );
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document.write( "I would very much appreciate some help here.\r<BR>\n" );
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document.write( "Thanks\r<BR>\n" );
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document.write( "Andreas\r<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 #2733 by </B><A HREF=/tutors/aboutme.mpl?userid=longjonsilver><B>longjonsilver(2297)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=longjonsilver><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=longjonsilver><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=2733\">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=1%2F%28log%28x%29%29 ALIGN=MIDDLE  ALT=\"1%2F%28log%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%28log%28sqrt%28x%2B2%29%29%29 ALIGN=MIDDLE  ALT=\"1%2F%28log%28sqrt%28x%2B2%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > -- cross multiply. Not sure if this changes the inequality round though!\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28log%28sqrt%28x%2B2%29%29%29 ALIGN=MIDDLE  ALT=\"%28log%28sqrt%28x%2B2%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28log%28x%29%29 ALIGN=MIDDLE  ALT=\"%28log%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > -- raise everything to power 2.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%2B2%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > -- now square both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2 ALIGN=MIDDLE  ALT=\"x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   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=x%5E2+-+x+-+2 ALIGN=MIDDLE  ALT=\"x%5E2+-+x+-+2\"   DISABLED_style= \"max-width:90%; height:auto;\" > >= 0\r<BR>\n" );
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document.write( "(x+1)(x-2) >= 0\r<BR>\n" );
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document.write( "So, this is asking where is the quadratic greater or equal than zero ie greater or equal than the x-axis (y=0). Well, we know the roots are now x=-1 and x=+2, so the answer is x<=-1 or x>=2: assuming that the inequality doesn't swap round when we cross multiplies the logs in the first step.\r<BR>\n" );
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document.write( "Is that clear :-)?\r<BR>\n" );
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document.write( "jon.<BR>\n" );
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