document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1026231>Question 1026231</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Graph the following system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the function for this region.\r<BR>\n" );
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document.write( "1. x &#8804; 2<BR>\n" );
document.write( "3x ­ -y &#8805; - ­2<BR>\n" );
document.write( "y &#8805; x ­ - 2<BR>\n" );
document.write( "f(x, y) = 2x - ­ 3y\r<BR>\n" );
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document.write( "2. g &#8804; - ­3h + 4<BR>\n" );
document.write( "g &#8805; ­3h - ­ 6<BR>\n" );
document.write( "g &#8805; (&#8531;)h- ­ 6<BR>\n" );
document.write( "f(g, h) = 2g ­ - 3h </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #642072 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=642072\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1. Question 1026330<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C=2 ALIGN=MIDDLE  ALT=\"x%3C=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x-y%3E=-2 ALIGN=MIDDLE  ALT=\"3x-y%3E=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B2%3E=y ALIGN=MIDDLE  ALT=\"3x%2B2%3E=y\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3E=x-2 ALIGN=MIDDLE  ALT=\"y%3E=x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "We can graph the boundary lines,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28x=2%29 ALIGN=MIDDLE  ALT=\"red%28x=2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28y=3x%2B2%29 ALIGN=MIDDLE  ALT=\"blue%28y=3x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28y=x-2%29 ALIGN=MIDDLE  ALT=\"green%28y=x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28200%2C300%2C-5%2C5%2C-5%2C10%2C%0D%0Agrid%281%29%2Cred%28line%282%2C-5%2C2%2C10%29%29%2C%0D%0Ablue%28line%28-3%2C-7%2C3%2C11%29%29%2C%0D%0Agreen%28line%28-3%2C-5%2C5%2C3%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We can find their intersections points (the vertices of the feasible region)<BR>\n" );
document.write( "by solving each 2-line system of equations:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=2%2Cy=x-2%29 ALIGN=MIDDLE  ALT=\"system%28x=2%2Cy=x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28system%28x=2%2Cy=0%29%29 ALIGN=MIDDLE  ALT=\"highlight%28system%28x=2%2Cy=0%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=2%2Cy=3x%2B2%29 ALIGN=MIDDLE  ALT=\"system%28x=2%2Cy=3x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28system%28x=2%2Cy=8%29%29 ALIGN=MIDDLE  ALT=\"highlight%28system%28x=2%2Cy=8%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y=3x%2B2%2Cy=x-2%29 ALIGN=MIDDLE  ALT=\"system%28y=3x%2B2%2Cy=x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28system%28x=-2%2Cy=-4%29%29 ALIGN=MIDDLE  ALT=\"highlight%28system%28x=-2%2Cy=-4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The function will be minimum at one of those vertices or all along the boundary line that joins two of them.<BR>\n" );
document.write( "Same thing goes for the maximum.<BR>\n" );
document.write( "So, we calculate the value for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%2C+y%29+=+2x-3y ALIGN=MIDDLE  ALT=\"f%28x%2C+y%29+=+2x-3y\"   DISABLED_style= \"max-width:90%; height:auto;\" > ate each one of those vertices:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%282%2C+0%29+=+2%2A2-3%2A0=4 ALIGN=MIDDLE  ALT=\"f%282%2C+0%29+=+2%2A2-3%2A0=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%282%2C+8%29+=+2%2A2-3%2A8=4-24=-20 ALIGN=MIDDLE  ALT=\"f%282%2C+8%29+=+2%2A2-3%2A8=4-24=-20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-2%2C+-4%29+=+2%28-2%29-3%28-4%29=-4%2B12=8 ALIGN=MIDDLE  ALT=\"f%28-2%2C+-4%29+=+2%28-2%29-3%28-4%29=-4%2B12=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The minimum is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28-20%29 ALIGN=MIDDLE  ALT=\"highlight%28-20%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The maximum is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%288%29 ALIGN=MIDDLE  ALT=\"highlight%288%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "2. I think one of the inequality signs was drawn backwards.<BR>\n" );
document.write( "The boundary lines are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28g=-3h%2B4%29 ALIGN=MIDDLE  ALT=\"red%28g=-3h%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28g=3h-6%29 ALIGN=MIDDLE  ALT=\"blue%28g=3h-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28g=%281%2F3%29h-6%29 ALIGN=MIDDLE  ALT=\"green%28g=%281%2F3%29h-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28200%2C300%2C-4%2C6%2C-8%2C7%2C%0D%0Agrid%281%29%2Cred%28line%28-2%2C10%2C5%2C-11%29%29%2C%0D%0Ablue%28line%28-2%2C-12%2C5%2C9%29%29%2C%0D%0Agreen%28line%28-6%2C-8%2C6%2C-4%29%29%2C%0D%0Alocate%280.4%2C6.9%2Cred%28g%29%29%2Clocate%285.5%2C1%2Cred%28h%29%29%2C%0D%0Ared%28line%285.3%2C-0.5%2C6%2C-0.5%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28g%3C=-3h%2B4%2Cg%3C=3h-6%2Cg%3E=%281%2F3%29h-6%29 ALIGN=MIDDLE  ALT=\"system%28g%3C=-3h%2B4%2Cg%3C=3h-6%2Cg%3E=%281%2F3%29h-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > represents the region at or below the red and blue lines, but at or above the green line.<BR>\n" );
document.write( "That region is the triangle bound by those lines (including the triangle sides).<BR>\n" );
document.write( "The vertices of that triangle are<BR>\n" );
document.write( "(0,-6) , (5/3,-1) and (3,-5). </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );