document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1120318>Question 1120318</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 2/x+1 < 4/4x-2 Fractional Inequality  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #735978 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13209)</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=735978\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Presumably the inequality is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F%28x%2B1%29%3C4%2F%284x-2%29 ALIGN=MIDDLE  ALT=\"2%2F%28x%2B1%29%3C4%2F%284x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F%28x%2B1%29%3C2%2F%282x-1%29 ALIGN=MIDDLE  ALT=\"2%2F%28x%2B1%29%3C2%2F%282x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Rewrite the inequality with \"0\" on one side and do a sign analysis on the resulting expression.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F%28x%2B1%29-2%2F%282x-1%29+%3C+0 ALIGN=MIDDLE  ALT=\"2%2F%28x%2B1%29-2%2F%282x-1%29+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%282x-1%29-2%28x%2B1%29%29%2F%28%28x%2B1%29%282x-1%29%29+%3C+0 ALIGN=MIDDLE  ALT=\"%282%282x-1%29-2%28x%2B1%29%29%2F%28%28x%2B1%29%282x-1%29%29+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284x-2-2x-2%29%2F%28%28x%2B1%29%282x-1%29%29+%3C+0 ALIGN=MIDDLE  ALT=\"%284x-2-2x-2%29%2F%28%28x%2B1%29%282x-1%29%29+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x-4%29%2F%28%28x%2B1%29%282x-1%29%29+%3C+0 ALIGN=MIDDLE  ALT=\"%282x-4%29%2F%28%28x%2B1%29%282x-1%29%29+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29%2F%28%28x%2B1%29%282x-1%29%29+%3C+0 ALIGN=MIDDLE  ALT=\"%28x-2%29%2F%28%28x%2B1%29%282x-1%29%29+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The expression is equal to 0 at x = 2; it is undefined at x = -1 and x = 1/2.<br><BR>\n" );
document.write( "The expression is positive for large positive values of x, because all the factors will be positive.<br><BR>\n" );
document.write( "As we \"walk\" along the number line, the sign of the expression changes each time we pass a point where the numerator or denominator is zero -- i.e., where the expression is either zero or undefined.  So<br><BR>\n" );
document.write( "The expression is zero or positive (the inequality is not true) on the interval [2, infinity);<BR>\n" );
document.write( "The expression is negative (the inequality is true) on the interval (1/2,2);<BR>\n" );
document.write( "The expression is again positive (the inequality is not true) on the interval (-1,1/2); and<BR>\n" );
document.write( "The expression is again positive (the inequality is true) on the interval (-infinity, -1)<br><BR>\n" );
document.write( "Answer: The solution set for the inequality is (-infinity,-1) union (1/2,2)<br><BR>\n" );
document.write( "The solution can be verified with a graph of 2/(x+1) (red) and 2/(2x-1) (green):<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-3%2C3%2C-10%2C10%2C2%2F%28x%2B1%29%2C2%2F%282x-1%29%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-3%2C3%2C-10%2C10%2C2%2F%28x%2B1%29%2C2%2F%282x-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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