document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1041377>Question 1041377</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Show that, if x is real, the values of the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2-3x%2B1%29%2F%282x%5E2-3x%2B2%29 ALIGN=MIDDLE  ALT=\"%28x%5E2-3x%2B1%29%2F%282x%5E2-3x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > can only lie between -1 and 5/7. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #656393 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=656393\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The domain of the expression is the set of all real numbers.  (The discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac+=+-17 ALIGN=MIDDLE  ALT=\"b%5E2-4ac+=+-17\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the denominator implies that it itself  is never zero.)\r<BR>\n" );
document.write( "\n" );
document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28x%5E2-3x%2B1%29%2F%282x%5E2-3x%2B2%29 ALIGN=MIDDLE  ALT=\"y+=+%28x%5E2-3x%2B1%29%2F%282x%5E2-3x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "\n" );
document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%282x%5E2-3x%2B2%29+=+x%5E2-3x%2B1 ALIGN=MIDDLE  ALT=\"y%282x%5E2-3x%2B2%29+=+x%5E2-3x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > (after multiplication; note that the denominator is never zero)\r<BR>\n" );
document.write( "\n" );
document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2yx%5E2-3yx%2B2y+=+x%5E2-3x%2B1 ALIGN=MIDDLE  ALT=\"2yx%5E2-3yx%2B2y+=+x%5E2-3x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > <==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282y-1%29x%5E2%2B%283-3y%29x%2B%282y-1%29+=+0 ALIGN=MIDDLE  ALT=\"%282y-1%29x%5E2%2B%283-3y%29x%2B%282y-1%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "\n" );
document.write( "For x to have real values, the discriminant of this equation (with x as variable) must be greater than or equal to 0. \r<BR>\n" );
document.write( "\n" );
document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283-3y%29%5E2-4%282y-1%29%5E2+%3E=+0 ALIGN=MIDDLE  ALT=\"%283-3y%29%5E2-4%282y-1%29%5E2+%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > <==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-7y%5E2-2y%2B5+%3E=0 ALIGN=MIDDLE  ALT=\"-7y%5E2-2y%2B5+%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7y%5E2%2B2y-5+%3C=0 ALIGN=MIDDLE  ALT=\"7y%5E2%2B2y-5+%3C=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >,\r<BR>\n" );
document.write( "\n" );
document.write( "<==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%287y-5%29%28y%2B1%29+%3C=+0 ALIGN=MIDDLE  ALT=\"%287y-5%29%28y%2B1%29+%3C=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "The inequality is only true when y &#8712; [-1, 5/7].\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );