document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=138674>Question 138674</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the inequality <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B5%29%2F%282x%5E2-5x-3%29%3E=0 ALIGN=MIDDLE  ALT=\"%28x%2B5%29%2F%282x%5E2-5x-3%29%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #101189 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=101189\">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=%28x%2B5%29%2F%282x%5E2-5x-3%29%3E=0 ALIGN=MIDDLE  ALT=\"%28x%2B5%29%2F%282x%5E2-5x-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=2x%5E2-5x-3=0 ALIGN=MIDDLE  ALT=\"2x%5E2-5x-3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the denominator equal to zero\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-3%29%282x%2B1%29=0 ALIGN=MIDDLE  ALT=\"%28x-3%29%282x%2B1%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor the left side (note: if you need help with factoring, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/playground/change-this-name4450.solver>solver</a>)\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=x-3=0 ALIGN=MIDDLE  ALT=\"x-3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B1=0 ALIGN=MIDDLE  ALT=\"2x%2B1=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=x=3 ALIGN=MIDDLE  ALT=\"x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1%2F2 ALIGN=MIDDLE  ALT=\"x=-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >    Now solve for x in each case\r<BR>\n" );
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document.write( "So the vertical asymptotes are<BR>\n" );
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document.write( "These are the first two critical values\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B5%29%2F%282x%5E2-5x-3%29%3E=0 ALIGN=MIDDLE  ALT=\"%28x%2B5%29%2F%282x%5E2-5x-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=%28x%2B5%29%2F%282x%5E2-5x-3%29=0 ALIGN=MIDDLE  ALT=\"%28x%2B5%29%2F%282x%5E2-5x-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( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-5 ALIGN=MIDDLE  ALT=\"x=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Solve for x\r<BR>\n" );
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document.write( "So our critical values are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3 ALIGN=MIDDLE  ALT=\"x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1%2F2 ALIGN=MIDDLE  ALT=\"x=-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-5 ALIGN=MIDDLE  ALT=\"x=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Now set up a number line and plot the critical values on the number line\r<BR>\n" );
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document.write( "So let's pick some test points that are near the critical values and evaluate them.\r<BR>\n" );
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document.write( "Let's pick a test value that is less than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5 ALIGN=MIDDLE  ALT=\"-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice how it's to the left of the leftmost endpoint):\r<BR>\n" );
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document.write( "So let's pick <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-6 ALIGN=MIDDLE  ALT=\"x=-6\"   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=%28x%2B5%29%2F%282x%5E2-5x-3%29%3E=0 ALIGN=MIDDLE  ALT=\"%28x%2B5%29%2F%282x%5E2-5x-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=%28-6%2B5%29%2F%282%28-6%29%5E2-5%28-6%29-3%29%3E=+0 ALIGN=MIDDLE  ALT=\"%28-6%2B5%29%2F%282%28-6%29%5E2-5%28-6%29-3%29%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-6 ALIGN=MIDDLE  ALT=\"x=-6\"   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=%28-1%29%2F99%3E=+0 ALIGN=MIDDLE  ALT=\"%28-1%29%2F99%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Evaluate and simplify the left side\r<BR>\n" );
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document.write( "Since the inequality is false, this means that the interval does <b>not</b> work. So this interval is <b>not</b> in our solution set and we can ignore it.\r<BR>\n" );
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document.write( "Let's pick a test value that is in between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5 ALIGN=MIDDLE  ALT=\"-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F2 ALIGN=MIDDLE  ALT=\"-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "So let's pick <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2 ALIGN=MIDDLE  ALT=\"x=-2\"   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=%28x%2B5%29%2F%282x%5E2-5x-3%29%3E=0 ALIGN=MIDDLE  ALT=\"%28x%2B5%29%2F%282x%5E2-5x-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=%28-2%2B5%29%2F%282%28-2%29%5E2%2B-5%28-2%29-3%29%3E=+0 ALIGN=MIDDLE  ALT=\"%28-2%2B5%29%2F%282%28-2%29%5E2%2B-5%28-2%29-3%29%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2 ALIGN=MIDDLE  ALT=\"x=-2\"   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=1%2F5%3E=+0 ALIGN=MIDDLE  ALT=\"1%2F5%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Evaluate and simplify the left side\r<BR>\n" );
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document.write( "Since the inequality is true, this means that the interval works. So this tells us that this interval is in our solution set.\r<BR>\n" );
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document.write( "   So part our solution in interval notation is <font size=\"8\">[</font><IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE -5,-\frac{1}{2}\"><font size=\"8\">)</font>\r<BR>\n" );
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document.write( "Let's pick a test value that is in between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F2 ALIGN=MIDDLE  ALT=\"-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "So let's pick <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   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=%28x%2B5%29%2F%282x%5E2-5x-3%29%3E=0 ALIGN=MIDDLE  ALT=\"%28x%2B5%29%2F%282x%5E2-5x-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=%281%2B5%29%2F%282%281%29%5E2-5%281%29-3%29%3E=+0 ALIGN=MIDDLE  ALT=\"%281%2B5%29%2F%282%281%29%5E2-5%281%29-3%29%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   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=-1%3E=+0 ALIGN=MIDDLE  ALT=\"-1%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Evaluate and simplify the left side\r<BR>\n" );
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document.write( "Since the inequality is false, this means that the interval does <b>not</b> work. So this interval is <b>not</b> in our solution set and we can ignore it.\r<BR>\n" );
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document.write( "Let's pick a test value that is greater than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice how it's to the right of the rightmost endpoint):\r<BR>\n" );
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document.write( "So let's pick <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   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=%28x%2B5%29%2F%282x%5E2-5x-3%29%3E=0 ALIGN=MIDDLE  ALT=\"%28x%2B5%29%2F%282x%5E2-5x-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=%284%2B5%29%2F%282%284%29%5E2-5%284%29-3%29%3E=+0 ALIGN=MIDDLE  ALT=\"%284%2B5%29%2F%282%284%29%5E2-5%284%29-3%29%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   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=1%3E=+0 ALIGN=MIDDLE  ALT=\"1%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Evaluate and simplify the left side\r<BR>\n" );
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document.write( "Since the inequality is true, this means that the interval works. So this tells us that this interval is in our solution set.\r<BR>\n" );
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document.write( "   So part our solution in interval notation is <font size=\"8\">(</font><IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 3,\infty\"><font size=\"8\">)</font>\r<BR>\n" );
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document.write( "Summary:\r<BR>\n" );
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document.write( "So the solution in interval notation is:\r<BR>\n" );
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