document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1168379>Question 1168379</A>: <I><SPAN STYLE=\"word-break: break-all;\"> use the graphical method to maximize: z = x+y<BR>\n" );
document.write( "Subject to: x+2y<12, 2x+y<12, x>0, y>0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #792999 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=792999\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> z = x + y is your objective function.\r<BR>\n" );
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document.write( "subject to inequalities are your constraints.\r<BR>\n" );
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document.write( "using the desmos.com calculator, you would graph the opposite of the inequalities.\r<BR>\n" );
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document.write( "the area on the graph that is not shaded is the region of feasibility.\r<BR>\n" );
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document.write( "the corner points of this region are where your maximum values of the objective function will be.\r<BR>\n" );
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document.write( "the corner points are in (x,y) format.\r<BR>\n" );
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document.write( "evaluate x + y using the values at each corner point.\r<BR>\n" );
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document.write( "the corner point with the greatest value of x + y is your maximum value of x + y.\r<BR>\n" );
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document.write( "the graph looks like this.\r<BR>\n" );
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document.write( "your points are at (0,6), (4,4), (6,0), (0,0)\r<BR>\n" );
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document.write( "your values of x + y at these points are 6, 8, 6, 0\r<BR>\n" );
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document.write( "if the points are allowed to be on the lines, your maximum value would be 8.\r<BR>\n" );
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document.write( "because the points can't be on the lines, your maximum value will be some value less than 8.\r<BR>\n" );
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document.write( "that's still the maximum value because all the other values are less than a maximum number that is less than 8.\r<BR>\n" );
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document.write( "fyi, i put this in a simplex modeling tool that is used to solve problems like this and it said it can't deal with <; it can only deal with <=.\r<BR>\n" );
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document.write( "i graphed the inequalities as <= or >=.\r<BR>\n" );
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document.write( "the corner points are the same as when i graphed the inequalities as < or >.\r<BR>\n" );
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document.write( "it's the evaluation at those corner points that counts.\r<BR>\n" );
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document.write( "if all inequalities were <=, then the value at the corner point is valid.\r<BR>\n" );
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document.write( "if some or all the inequalities were <, then the value at the corner point needs to be checked to see that all the inequalities are satisfied at that corner point.\r<BR>\n" );
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document.write( "the simplex modeling tool i used is at:\r<BR>\n" );
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document.write( "https://www.zweigmedia.com/RealWorld/simplex.html\r<BR>\n" );
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document.write( "the desmos.com graphing calculator that i used is at:\r<BR>\n" );
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document.write( "https://www.desmos.com/calculator\r<BR>\n" );
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document.write( "here's a display of the graphing method in the two forms described above (using <= and using <) to show you that the corner points are the same.\r<BR>\n" );
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document.write( "here's a display of the results using the simplex modeling tool in the two forms describe above to show you that the simplex modeling tool could handle <= and couldn't can't handle <.\r<BR>\n" );
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