document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1189461>Question 1189461</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Answer the following logarithmic inequalities.\r<BR>\n" );
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document.write( "1) log3(1-x)>=log3(x+16-x^2)\r<BR>\n" );
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document.write( "(Note: 3 is a base)<BR>\n" );
document.write( "Thank you. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #820855 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=820855\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%283%2C%281-x%29%29%3E=log%283%2C%28-x%5E2%2Bx%2B16%29%29 ALIGN=MIDDLE  ALT=\"log%283%2C%281-x%29%29%3E=log%283%2C%28-x%5E2%2Bx%2B16%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "To start with, both the arguments must be positive:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281-x%29%3E0 ALIGN=MIDDLE  ALT=\"%281-x%29%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C+1 ALIGN=MIDDLE  ALT=\"x+%3C+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-x%5E2%2Bx%2B16%29%3E0 ALIGN=MIDDLE  ALT=\"%28-x%5E2%2Bx%2B16%29%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3.53+%3C+x+%3C+4.53 ALIGN=MIDDLE  ALT=\"-3.53+%3C+x+%3C+4.53\"   DISABLED_style= \"max-width:90%; height:auto;\" > (approximately)<br><BR>\n" );
document.write( "Putting those together, the possible approximate range of solutions is<br><BR>\n" );
document.write( "[1] <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3.53+%3C+x+%3C+5.11 ALIGN=MIDDLE  ALT=\"-3.53+%3C+x+%3C+5.11\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "Now we need to find which values of x in that range satisfy the given inequality.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%283%2C%281-x%29%29%3E=log%283%2C%28-x%5E2%2Bx%2B16%29%29 ALIGN=MIDDLE  ALT=\"log%283%2C%281-x%29%29%3E=log%283%2C%28-x%5E2%2Bx%2B16%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Since log(A) is monotonic...<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-x%3E=-x%5E2%2Bx%2B16 ALIGN=MIDDLE  ALT=\"1-x%3E=-x%5E2%2Bx%2B16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-2x-15%3E=0 ALIGN=MIDDLE  ALT=\"x%5E2-2x-15%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-5%29%28x%2B3%29%3E=0 ALIGN=MIDDLE  ALT=\"%28x-5%29%28x%2B3%29%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "[2] <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E5 ALIGN=MIDDLE  ALT=\"x%3E5\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C-3 ALIGN=MIDDLE  ALT=\"x%3C-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Combining [1] and [2], we see that the inequality is satisfied only on the interval from -3.53 (approximately) to -3.<br><BR>\n" );
document.write( "ANSWER: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3.53%3Cx%3C-3 ALIGN=MIDDLE  ALT=\"-3.53%3Cx%3C-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( " </SPAN>\n" );
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