document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=306988>Question 306988</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What happens if in a rational inequality the numerator is imaginary? This problem was on our test and now I need help for finals:<BR>\n" );
document.write( "Let f(x)=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B3%29%2F%28x%2B1%29 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%2F%28x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and g(x)=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F%28x%2B4%29 ALIGN=MIDDLE  ALT=\"2%2F%28x%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. For what values of x is the graph of y=f(x) below the graph of y=g(x)?<BR>\n" );
document.write( "I didn't know exactly what to do, but I set f(x) less than g(x) and tried to solve for the solutions, but the numerator is imaginary and I don't know what to do. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #219660 by </B><A HREF=/tutors/aboutme.mpl?userid=Fombitz><B>Fombitz(32388)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fombitz><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=Fombitz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=219660\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Imaginary, what do you mean? <BR>\n" );
document.write( "You can solve algebraically or by graphing.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-10%2C8%2C-8%2C8%2Cblue%28line%28-1%2C-10%2C-1%2C10%29%29%2Cblue%28line%28-4%2C-10%2C-4%2C10%29%29%2C+graph%28+300%2C+300%2C+-10%2C+8%2C+-8%2C8%2C+%28x%2B3%29%2F%28x%2B1%29%2C+2%2F%28x%2B4%29%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "f(x) is the red function.<BR>\n" );
document.write( "g(x) is the green function.<BR>\n" );
document.write( "You can clearly make out that between (-4,-1) that f(x) is less than g(x). <BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Now let's show it algebraically.<BR>\n" );
document.write( "Both function have singularities. <BR>\n" );
document.write( "f(x) at x=-1 and g(x) at x=-4.<BR>\n" );
document.write( "Break up the number line into three regions and check the inequality in those regions. You can pick any point in the region just not the end points. <BR>\n" );
document.write( "1:(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-infinity ALIGN=MIDDLE  ALT=\"-infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "2:(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "3:(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Region 1: Let x=-5.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B3%29%2F%28x%2B1%29%3C2%2F%28x%2B4%29 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%2F%28x%2B1%29%3C2%2F%28x%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-5%2B3%29%2F%28-5%2B1%29%3C2%2F%28-5%2B4%29 ALIGN=MIDDLE  ALT=\"%28-5%2B3%29%2F%28-5%2B1%29%3C2%2F%28-5%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-2%29%2F%28-4%29%3C-2 ALIGN=MIDDLE  ALT=\"%28-2%29%2F%28-4%29%3C-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2%3C-2 ALIGN=MIDDLE  ALT=\"1%2F2%3C-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "No, not a valid region.<BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Region 2: Let x=-2.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B3%29%2F%28x%2B1%29%3C2%2F%28x%2B4%29 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%2F%28x%2B1%29%3C2%2F%28x%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-2%2B3%29%2F%28-2%2B1%29%3C2%2F%28-2%2B4%29 ALIGN=MIDDLE  ALT=\"%28-2%2B3%29%2F%28-2%2B1%29%3C2%2F%28-2%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%29%2F%28-1%29%3C2%2F%282%29 ALIGN=MIDDLE  ALT=\"%281%29%2F%28-1%29%3C2%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%3C1 ALIGN=MIDDLE  ALT=\"-1%3C1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Yes, a valid region.<BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Region 3: Let x=0.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B3%29%2F%28x%2B1%29%3C2%2F%28x%2B4%29 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%2F%28x%2B1%29%3C2%2F%28x%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%29%2F%281%29%3C2%2F%284%29 ALIGN=MIDDLE  ALT=\"%283%29%2F%281%29%3C2%2F%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%3C1%2F2 ALIGN=MIDDLE  ALT=\"3%3C1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "No, not a valid region.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );