document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=30039>Question 30039</A>: <I><SPAN STYLE=\"word-break: break-all;\"> x-2/x+6 >= 3<BR>\n" );
document.write( "Please help me solve this equation </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #16775 by </B><A HREF=/tutors/aboutme.mpl?userid=Fermat><B>Fermat(136)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fermat><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=Fermat><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=16775\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Multiply everything by x<BR>\n" );
document.write( "giving,<BR>\n" );
document.write( "x² - 2 + 6x >= 3x (assuming x is positive)<BR>\n" );
document.write( "rearrange,<BR>\n" );
document.write( "x² + 3x - 2 >= 0<BR>\n" );
document.write( "use the quadratic formula,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "a = 1, b = 3, c = -2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-3+%2B-+sqrt%28+3%5E2-4%2A1%2A-2+%29%29%2F%282%2A1%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-3+%2B-+sqrt%28+3%5E2-4%2A1%2A-2+%29%29%2F%282%2A1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-3+%2B-+sqrt%28+17+%29%29%2F%282%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-3+%2B-+sqrt%28+17+%29%29%2F%282%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "x = 0.5616, x = -3.5616\r<BR>\n" );
document.write( "\n" );
document.write( "(x-0.5616)(x+3.5616) >= 0<BR>\n" );
document.write( "=========================<BR>\n" );
document.write( "Since the product of the two terms is positive (>=0), then the terms must be both positive or both negative (negative times negative = positive).<BR>\n" );
document.write( "So,<BR>\n" );
document.write( "x >= 0.5616 and x >= -3.5616<BR>\n" );
document.write( "i.e. x >= 0.5616<BR>\n" );
document.write( "================<BR>\n" );
document.write( "or<BR>\n" );
document.write( "x =< 0.5616 and x =< -3.5616<BR>\n" );
document.write( "i.e. x =< -3.5616<BR>\n" );
document.write( "================\r<BR>\n" );
document.write( "\n" );
document.write( "But x was assumed to be positive (see third line). So the last solution (x =< -3.5616) is invalid since it requires x to be negative)<BR>\n" );
document.write( "In fact, if you assume x to be negative, you get,<BR>\n" );
document.write( "x² - 2 + 6x <= 3x (notice the change in the inequality sign)<BR>\n" );
document.write( "rearrange,<BR>\n" );
document.write( "x² + 3x - 2 <= 0<BR>\n" );
document.write( "giving,<BR>\n" );
document.write( "(x-0.5616)(x+3.5616) <= 0<BR>\n" );
document.write( "==========================<BR>\n" );
document.write( "This requires one term to be positive and the other term to be negative - at the same time.<BR>\n" );
document.write( "So,<BR>\n" );
document.write( "x =< 0.5616 and x >= -3.5616<BR>\n" );
document.write( "or<BR>\n" );
document.write( "-3.5616 =< x =< 0.5616<BR>\n" );
document.write( "but x requires to be negative, so this becomes,<BR>\n" );
document.write( "-3.5616 =< x =< 0<BR>\n" );
document.write( "=================<BR>\n" );
document.write( "The two solutions are,<BR>\n" );
document.write( "x >= 0.5616<BR>\n" );
document.write( "===========<BR>\n" );
document.write( "and<BR>\n" );
document.write( "-3.5616 =< x =< 0<BR>\n" );
document.write( "================= </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );