document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1019700>Question 1019700</A>: <I><SPAN STYLE=\"word-break: break-all;\"> <I am a tutor, and am a bit confused by the answer sheet provided to my student, by their teacher. I might be wrong =) but I am wondering if the answer sheet is wrong.<BR>\n" );
document.write( "SOLVE THE INEQUALITY ALGEBRAICALLY:<BR>\n" );
document.write( "2/X+1 >= 1/X-2<BR>\n" );
document.write( "I rearranged this to: 5-x/(x-2)(x+1)<=0  <BR>\n" );
document.write( "The teacher's solution/answer states (-1,2) U (6,infinity)<BR>\n" );
document.write( "I got 5 instead of 6.  ???<BR>\n" );
document.write( "HELP PLEASE =) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #635650 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=635650\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i graphed both equation and this is what i got.\r<BR>\n" );
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document.write( "so, it appears that you are right.\r<BR>\n" );
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document.write( "it should be greater than or equal to 5.\r<BR>\n" );
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document.write( "the interval would show as [5,infinity).\r<BR>\n" );
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document.write( "since it was greater than or equal, rather than just greater than, x = 5 is a good solution as well as everything to the right of x = 5.\r<BR>\n" );
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document.write( "the first graph shows each equation individually.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2016/021402.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "the second graph shows the first expression minus the second expression.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2016/021403.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "so the graphical solution shows you're right.\r<BR>\n" );
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document.write( "i had a little more trouble with the algebraic solution, but i think i finally figured it out.\r<BR>\n" );
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document.write( "these problems can be tricky and it always takes me more than one try to get them.\r<BR>\n" );
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document.write( "start with 2/(x+1) >= 1/(x-2)\r<BR>\n" );
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document.write( "subtract 1/(x+2) from both sides of the equation to get:\r<BR>\n" );
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document.write( "2/(x+1) - (/x-2) >= 0\r<BR>\n" );
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document.write( "put everything under a common denominator to get:\r<BR>\n" );
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document.write( "(2*(x-2) - (x+1) / ((x+1)*(x-2)) >= 0\r<BR>\n" );
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document.write( "simplify to get (2x - 4 - x - 1) / ((x+1)*(x-2)) >= 0\r<BR>\n" );
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document.write( "simplify further to get (x-5) / ((x+1)*(x-2)) >= 0\r<BR>\n" );
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document.write( "since (x-5) = 0 when x = 5, it's clear that x = 5 has to be in the solution set.\r<BR>\n" );
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document.write( "it is also clear that x cannot be -1 or 2 because then you would have a division by 0.\r<BR>\n" );
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document.write( "so there are 3 points to graph and find the sign of the equation between.\r<BR>\n" );
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document.write( "they are x = -1, x = 2, and x = 5.\r<BR>\n" );
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document.write( "there are vertical asymptotes at x = -1 and x = 2\r<BR>\n" );
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document.write( "all that's left is to figure out whether the equation is positive or negative in between the intervals shown.\r<BR>\n" );
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document.write( "when x = -2 (smaller than -1), then the equaation is equal to -1.75 which is negative.\r<BR>\n" );
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document.write( "when x = 7 (greater than 5), then the equation is equal to .05 which is positive.\r<BR>\n" );
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document.write( "when x = 0 (between -1 and 2), then the equation is equal to 2.5 which is positive.\r<BR>\n" );
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document.write( "i think i did the math correctly.\r<BR>\n" );
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document.write( "the algebra confirms the graph and vice versa so the solution looks good.\r<BR>\n" );
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document.write( "i've seen several instances where the answer sheet is wrong, so it's not to be unexpected.\r<BR>\n" );
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document.write( "just the fact that the equation was equal to 0 when x = 5 proves you were right.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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