document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=86136>Question 86136</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need some help with these graph problems.  I would appreciate some help with solving and graphing these systems. Thank you.\r<BR>\n" );
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document.write( "Solve the system by graphing.<BR>\n" );
document.write( "x+y=4<BR>\n" );
document.write( "-x+y=2\r<BR>\n" );
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document.write( "Solve the following systym of linear inequalities by graphing.<BR>\n" );
document.write( "3x+4y<12<BR>\n" );
document.write( "x+3y<6<BR>\n" );
document.write( "x>0<BR>\n" );
document.write( "y>0 (under each of the greater and lesser signs is a line)\r<BR>\n" );
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document.write( "Solve the following system of linear inequalities by graphing.<BR>\n" );
document.write( "x+2y<3<BR>\n" );
document.write( "2x-3y<6\r<BR>\n" );
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document.write( "Solve the following system of linear inequalities by graphing<BR>\n" );
document.write( "x-2y>4<BR>\n" );
document.write( "x<4\r<BR>\n" );
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document.write( "Thank you so much for taking your time to help me out. thanks alot. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #62285 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=62285\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=solve_by_graphing.solver>Solve the System of Equations by Graphing</A></TD></TR>\n" );
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document.write( "  Start with the given system of equations:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%2By=4 ALIGN=MIDDLE  ALT=\"1x%2By=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x%2By=2 ALIGN=MIDDLE  ALT=\"-x%2By=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  In order to graph these equations, we need to solve for y for each equation.<BR>\n" );
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document.write( "  So let's solve for y on the first equation<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%2By=4 ALIGN=MIDDLE  ALT=\"1x%2By=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1y=4-x ALIGN=MIDDLE  ALT=\"1y=4-x\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x ALIGN=MIDDLE  ALT=\"+x\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1y=-x%2B4 ALIGN=MIDDLE  ALT=\"1y=-x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the equation<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-x%2B4%29%2F%281%29 ALIGN=MIDDLE  ALT=\"y=%28-x%2B4%29%2F%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%2F1%29x%2B%284%29%2F%281%29 ALIGN=MIDDLE  ALT=\"y=%28-1%2F1%29x%2B%284%29%2F%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the fraction<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-x%2B4 ALIGN=MIDDLE  ALT=\"y=-x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce<BR>\n" );
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document.write( "  Now lets graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-x%2B4 ALIGN=MIDDLE  ALT=\"y=-x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > (note: if you need help with graphing, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Linear-equations/graphing-linear-equations.solver>solver</a>)<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+-x%2B4%29+ ALIGN=MIDDLE  ALT=\"+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+-x%2B4%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-x%2B4 ALIGN=MIDDLE  ALT=\"y=-x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  So let's solve for y on the second equation<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x%2By=2 ALIGN=MIDDLE  ALT=\"-x%2By=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1y=2%2Bx ALIGN=MIDDLE  ALT=\"1y=2%2Bx\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x ALIGN=MIDDLE  ALT=\"+x\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both sides<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1y=%2Bx%2B2 ALIGN=MIDDLE  ALT=\"1y=%2Bx%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the equation<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28%2Bx%2B2%29%2F%281%29 ALIGN=MIDDLE  ALT=\"y=%28%2Bx%2B2%29%2F%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28%2B1%2F1%29x%2B%282%29%2F%281%29 ALIGN=MIDDLE  ALT=\"y=%28%2B1%2F1%29x%2B%282%29%2F%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the fraction<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%2B2 ALIGN=MIDDLE  ALT=\"y=x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce<BR>\n" );
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document.write( "  Now lets add the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%2B2 ALIGN=MIDDLE  ALT=\"y=x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to our first plot to get:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+-x%2B4%2Cx%2B2%29+ ALIGN=MIDDLE  ALT=\"+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+-x%2B4%2Cx%2B2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-x%2B4 ALIGN=MIDDLE  ALT=\"y=-x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" >(red) and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%2B2 ALIGN=MIDDLE  ALT=\"y=x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" >(green)<BR>\n" );
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document.write( "  From the graph, we can see that the two lines intersect at the point (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >) (note: you might have to adjust the window to see the intersection)</TD></TR>\n" );
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document.write( "--------------------------------------------------------------------------\r<BR>\n" );
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document.write( "Start with the given system of inequalities\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B4y%3C=12 ALIGN=MIDDLE  ALT=\"3x%2B4y%3C=12\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B3y%3C=6 ALIGN=MIDDLE  ALT=\"x%2B3y%3C=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E=0 ALIGN=MIDDLE  ALT=\"x%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3E=0 ALIGN=MIDDLE  ALT=\"y%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "In order to graph this system of inequalities, we need to graph each inequality one at a time.\r<BR>\n" );
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document.write( "So lets graph the first inequality\r<BR>\n" );
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document.write( "    In order to graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B4y%3C=12 ALIGN=MIDDLE  ALT=\"3x%2B4y%3C=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > we need to graph the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B4y=12 ALIGN=MIDDLE  ALT=\"3x%2B4y=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > (just replace the inequality sign with an equal sign). So lets graph the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B4y=12 ALIGN=MIDDLE  ALT=\"3x%2B4y=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > (note: if you need help with graphing, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Linear-equations/graphing-linear-equations.solver>solver<a>)\r<BR>\n" );
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document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%29 ALIGN=MIDDLE  ALT=\"graph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B4y=12 ALIGN=MIDDLE  ALT=\"3x%2B4y=12\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "    Now lets pick a test point, say (0,0) (any point will work, but this point is the easiest to work with), and evaluate the inequality <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B4y%3C=12 ALIGN=MIDDLE  ALT=\"3x%2B4y%3C=12\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "     <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%280%29%2B4%280%29%3C=12 ALIGN=MIDDLE  ALT=\"3%280%29%2B4%280%29%3C=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in x=0, y=0\r<BR>\n" );
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document.write( "     <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C=12 ALIGN=MIDDLE  ALT=\"0%3C=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify\r<BR>\n" );
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document.write( "Since this inequality is true, we shade the entire region containing (0,0)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2Cgraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-0%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-0.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-1.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-2.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-3%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-3.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-4.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-5.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-6%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-6.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-7.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-8.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-9%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-9.75%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " Here is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B4y%3C=12 ALIGN=MIDDLE  ALT=\"3x%2B4y%3C=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > with the graph of the line(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B4y=12 ALIGN=MIDDLE  ALT=\"3x%2B4y=12\"   DISABLED_style= \"max-width:90%; height:auto;\" >) in red and the shaded region in green\r<BR>\n" );
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document.write( "(note: In this case, the red line is a solid line. This means the boundaries are included in the region.)\r<BR>\n" );
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document.write( "Now lets graph the second inequality\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "    In order to graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B3y%3C=6 ALIGN=MIDDLE  ALT=\"x%2B3y%3C=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > we need to graph the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B3y=6 ALIGN=MIDDLE  ALT=\"x%2B3y=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > (just replace the inequality sign with an equal sign). So lets graph the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B3y=6 ALIGN=MIDDLE  ALT=\"x%2B3y=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > (note: if you need help with graphing, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Linear-equations/graphing-linear-equations.solver>solver<a>)\r<BR>\n" );
document.write( "\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%29 ALIGN=MIDDLE  ALT=\"graph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B3y=6 ALIGN=MIDDLE  ALT=\"x%2B3y=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "    Now lets pick a test point, say (0,0) (any point will work, but this point is the easiest to work with), and evaluate the inequality <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B3y%3C=6 ALIGN=MIDDLE  ALT=\"x%2B3y%3C=6\"   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=%280%29%2B3%280%29%3C=6 ALIGN=MIDDLE  ALT=\"%280%29%2B3%280%29%3C=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in x=0, y=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "     <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C=6 ALIGN=MIDDLE  ALT=\"0%3C=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since this inequality is true, we shade the entire region containing (0,0)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2Cgraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-0%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-0.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-1.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-2.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-3%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-3.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-4.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-5.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-6%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-6.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-7.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-8.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-9%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-9.75%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( " Here is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B3y%3C=6 ALIGN=MIDDLE  ALT=\"x%2B3y%3C=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > with the graph of the line(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B3y=6 ALIGN=MIDDLE  ALT=\"x%2B3y=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >) in red and the shaded region in green\r<BR>\n" );
document.write( "\n" );
document.write( "(note: In this case, the red line is a solid line. This means the boundaries are included in the region.)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now lets graph the third inequality\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "    In order to graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E=0 ALIGN=MIDDLE  ALT=\"x%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > we need to graph the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (just replace the inequality sign with an equal sign). So lets graph the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (note: if you need help with graphing, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Linear-equations/graphing-linear-equations.solver>solver<a>)\r<BR>\n" );
document.write( "\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C1000%28x-100%2F1000%29%29 ALIGN=MIDDLE  ALT=\"graph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C1000%28x-100%2F1000%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (it is the same line as the axis)\r<BR>\n" );
document.write( "\n" );
document.write( "    Now lets pick a test point, say (1,0) (any point will work, but this point is the easiest to work with), and evaluate the inequality <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E=0 ALIGN=MIDDLE  ALT=\"x%3E=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=%281%29%3E=0 ALIGN=MIDDLE  ALT=\"%281%29%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in x=1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "     <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%3E=0 ALIGN=MIDDLE  ALT=\"1%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since this inequality is true, we shade the entire region that contain (1,0)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2Cgraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-0%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-0.75%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-1.5%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-2.25%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-3%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-3.75%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-4.5%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-5.25%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-6%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-6.75%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-7.5%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-8.25%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-9%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-9.75%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( " Here is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E=0 ALIGN=MIDDLE  ALT=\"x%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > with the graph of the line(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) in red and the shaded region in green\r<BR>\n" );
document.write( "\n" );
document.write( "(note: In this case, the red line is a solid line. This means the boundaries are included in the region.)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now lets graph the fourth inequality\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "    In order to graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3E=0 ALIGN=MIDDLE  ALT=\"y%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > we need to graph the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (just replace the inequality sign with an equal sign). So lets graph the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (note: if you need help with graphing, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Linear-equations/graphing-linear-equations.solver>solver<a>)\r<BR>\n" );
document.write( "\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%29 ALIGN=MIDDLE  ALT=\"graph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ((it is the same line as the axis)\r<BR>\n" );
document.write( "\n" );
document.write( "    Now lets pick a test point, say (0,1) (any point will work, but this point is the easiest to work with), and evaluate the inequality <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3E=0 ALIGN=MIDDLE  ALT=\"y%3E=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=%281%29%3E=0 ALIGN=MIDDLE  ALT=\"%281%29%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in y=1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "     <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%3E=0 ALIGN=MIDDLE  ALT=\"1%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since this inequality is true, we shade the entire region that contain (0,1)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2Cgraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B0%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B0.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B1.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B2.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B3%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B3.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B4.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B5.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B6%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B6.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B7.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B8.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B9%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B9.75%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( " Here is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3E=0 ALIGN=MIDDLE  ALT=\"y%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > with the graph of the line(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) in red and the shaded region in green\r<BR>\n" );
document.write( "\n" );
document.write( "(note: In this case, the red line is a solid line. This means the boundaries are included in the region.)\r<BR>\n" );
document.write( "\n" );
document.write( "So we essentially have these 4 regions\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2Cgraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-0%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-0.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-1.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-2.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-3%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-3.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-4.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-5.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-6%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-6.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-7.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-8.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-9%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-9.75%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Region #1 which is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B4y%3C=12 ALIGN=MIDDLE  ALT=\"3x%2B4y%3C=12\"   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=drawing%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2Cgraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-0%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-0.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-1.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-2.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-3%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-3.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-4.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-5.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-6%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-6.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-7.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-8.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-9%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-9.75%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Region #2 which is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B3y%3C=6 ALIGN=MIDDLE  ALT=\"x%2B3y%3C=6\"   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=drawing%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2Cgraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-0%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-0.75%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-1.5%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-2.25%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-3%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-3.75%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-4.5%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-5.25%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-6%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-6.75%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-7.5%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-8.25%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-9%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-9.75%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Region #3 which is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E=0 ALIGN=MIDDLE  ALT=\"x%3E=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=drawing%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2Cgraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B0%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B0.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B1.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B2.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B3%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B3.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B4.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B5.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B6%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B6.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B7.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B8.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B9%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B9.75%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Region #4 which is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3E=0 ALIGN=MIDDLE  ALT=\"y%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So these regions overlap to produce this region. It's a little hard to see, but after evenly shading each region, the intersecting region will be the most shaded in.\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0A%0D%0Adrawing%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C+-10%2C+10%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-0%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-0.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-1.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-2.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-3%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-3.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-4.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-5.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-6%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-6.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-7.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-8.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-9%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C+%2812-3%2Ax%29%2F4-9.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-0%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-0.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-1.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-2.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-3%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-3.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-4.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-5.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-6%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-6.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-7.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-8.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-9%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F3%2C+%286-1%2Ax%29%2F3-9.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-0%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-0.75%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-1.5%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-2.25%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-3%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-3.75%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-4.5%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-5.25%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-6%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-6.75%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-7.5%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-8.25%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-9%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-100%2F1000%29%2C+10000%28x-100%2F1000-9.75%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B0%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B0.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B1.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B2.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B3%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B3.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B4.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B5.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B6%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B6.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B7.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B8.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B9%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%28100-1%2Ax%29%2F1000%2C+%28100-1%2Ax%29%2F1000%2B9.75%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Here is a cleaner look at the intersection of regions\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%0D%0A++graph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%2812-3%2Ax%29%2F4%2C%286-1%2Ax%29%2F3%2C%28100-1000%2Ax%29%2F1%2C%28100-1%2Ax%29%2F1000%29%2Ccircle%280.5%2C0.5%2C0.05%29%2C%0D%0A+circle%280.5%2C0.5%2C0.08%29%2Ccircle%280.5%2C1%2C0.05%29%2C%0D%0A+circle%280.5%2C1%2C0.08%29%2Ccircle%280.5%2C1.5%2C0.05%29%2C%0D%0A+circle%280.5%2C1.5%2C0.08%29%2Ccircle%281%2C0.5%2C0.05%29%2C%0D%0A+circle%281%2C0.5%2C0.08%29%2Ccircle%281%2C1%2C0.05%29%2C%0D%0A+circle%281%2C1%2C0.08%29%2Ccircle%281%2C1.5%2C0.05%29%2C%0D%0A+circle%281%2C1.5%2C0.08%29%2Ccircle%281.5%2C0.5%2C0.05%29%2C%0D%0A+circle%281.5%2C0.5%2C0.08%29%2Ccircle%281.5%2C1%2C0.05%29%2C%0D%0A+circle%281.5%2C1%2C0.08%29%2Ccircle%282%2C0.5%2C0.05%29%2C%0D%0A+circle%282%2C0.5%2C0.08%29%2Ccircle%282%2C1%2C0.05%29%2C%0D%0A+circle%282%2C1%2C0.08%29%2Ccircle%282.5%2C0.5%2C0.05%29%2C%0D%0A+circle%282.5%2C0.5%2C0.08%29%2Ccircle%282.5%2C1%2C0.05%29%2C%0D%0A+circle%282.5%2C1%2C0.08%29%2Ccircle%283%2C0.5%2C0.05%29%2C%0D%0A+circle%283%2C0.5%2C0.08%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Here is the intersection of the 4 regions represented by the dots\r<BR>\n" );
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document.write( "--------------------------------------------------------------------------\r<BR>\n" );
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document.write( "Start with the given system of inequalities\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2y%3C=3 ALIGN=MIDDLE  ALT=\"x%2B2y%3C=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-3y%3C=6 ALIGN=MIDDLE  ALT=\"2x-3y%3C=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "In order to graph this system of inequalities, we need to graph each inequality one at a time.\r<BR>\n" );
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document.write( "So lets graph the first inequality\r<BR>\n" );
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document.write( "    In order to graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2y%3C=3 ALIGN=MIDDLE  ALT=\"x%2B2y%3C=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > we need to graph the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2y=3 ALIGN=MIDDLE  ALT=\"x%2B2y=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (just replace the inequality sign with an equal sign). So lets graph the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2y=3 ALIGN=MIDDLE  ALT=\"x%2B2y=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (note: if you need help with graphing, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Linear-equations/graphing-linear-equations.solver>solver<a>)\r<BR>\n" );
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document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%29 ALIGN=MIDDLE  ALT=\"graph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2y=3 ALIGN=MIDDLE  ALT=\"x%2B2y=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "    Now lets pick a test point, say (0,0) (any point will work, but this point is the easiest to work with), and evaluate the inequality <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2y%3C=3 ALIGN=MIDDLE  ALT=\"x%2B2y%3C=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "     <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%280%29%2B2%280%29%3C=3 ALIGN=MIDDLE  ALT=\"%280%29%2B2%280%29%3C=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in x=0, y=0\r<BR>\n" );
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document.write( "     <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C=3 ALIGN=MIDDLE  ALT=\"0%3C=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify\r<BR>\n" );
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document.write( "Since this inequality is true, we shade the entire region containing (0,0)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2Cgraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-0%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-0.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-1.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-2.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-3%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-3.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-4.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-5.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-6%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-6.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-7.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-8.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-9%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-9.75%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " Here is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2y%3C=3 ALIGN=MIDDLE  ALT=\"x%2B2y%3C=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > with the graph of the line(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2y=3 ALIGN=MIDDLE  ALT=\"x%2B2y=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >) in red and the shaded region in green\r<BR>\n" );
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document.write( "(note: In this case, the red line is a solid line. This means the boundaries are included in the region.)\r<BR>\n" );
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document.write( "Now lets graph the second inequality\r<BR>\n" );
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document.write( "    In order to graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-3y%3C=6 ALIGN=MIDDLE  ALT=\"2x-3y%3C=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > we need to graph the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-3y=6 ALIGN=MIDDLE  ALT=\"2x-3y=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > (just replace the inequality sign with an equal sign). So lets graph the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-3y=6 ALIGN=MIDDLE  ALT=\"2x-3y=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > (note: if you need help with graphing, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Linear-equations/graphing-linear-equations.solver>solver<a>)\r<BR>\n" );
document.write( "\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%29 ALIGN=MIDDLE  ALT=\"graph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-3y=6 ALIGN=MIDDLE  ALT=\"2x-3y=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "    Now lets pick a test point, say (0,0) (any point will work, but this point is the easiest to work with), and evaluate the inequality <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-3y%3C=6 ALIGN=MIDDLE  ALT=\"2x-3y%3C=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "     <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%280%29-3%280%29%3C=6 ALIGN=MIDDLE  ALT=\"2%280%29-3%280%29%3C=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in x=0, y=0\r<BR>\n" );
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document.write( "     <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C=6 ALIGN=MIDDLE  ALT=\"0%3C=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify\r<BR>\n" );
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document.write( "Since this inequality is true, we shade the entire region containing (0,0)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2Cgraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B0%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B0.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B1.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B2.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B3%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B3.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B4.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B5.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B6%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B6.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B7.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B8.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B9%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B9.75%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( " Here is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-3y%3C=6 ALIGN=MIDDLE  ALT=\"2x-3y%3C=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > with the graph of the line(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-3y=6 ALIGN=MIDDLE  ALT=\"2x-3y=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >) in red and the shaded region in green\r<BR>\n" );
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document.write( "(note: In this case, the red line is a solid line. This means the boundaries are included in the region.)\r<BR>\n" );
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document.write( "So we essentially have these 2 regions\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2Cgraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-0%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-0.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-1.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-2.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-3%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-3.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-4.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-5.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-6%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-6.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-7.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-8.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-9%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-9.75%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Region #1 which is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2y%3C=3 ALIGN=MIDDLE  ALT=\"x%2B2y%3C=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2Cgraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B0%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B0.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B1.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B2.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B3%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B3.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B4.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B5.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B6%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B6.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B7.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B8.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B9%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B9.75%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Region #2 which is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-3y%3C=6 ALIGN=MIDDLE  ALT=\"2x-3y%3C=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So these regions overlap to produce this region. It's a little hard to see, but after evenly shading each region, the intersecting region will be the most shaded in.\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0A%0D%0Adrawing%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C+-10%2C+10%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-0%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-0.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-1.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-2.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-3%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-3.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-4.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-5.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-6%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-6.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-7.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-8.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-9%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%2C+%283-1%2Ax%29%2F2-9.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B0%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B0.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B1.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B2.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B3%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B3.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B4.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B5.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B6%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B6.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B7.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B8.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B9%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%2C+%286-2%2Ax%29%2F-3%2B9.75%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Here is a cleaner look at the intersection of regions\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
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ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Here is the intersection of the 2 regions represented by the dots\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "--------------------------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Start with the given system of inequalities\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-2y%3E=4 ALIGN=MIDDLE  ALT=\"x-2y%3E=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C=4 ALIGN=MIDDLE  ALT=\"x%3C=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "In order to graph this system of inequalities, we need to graph each inequality one at a time.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So lets graph the first inequality\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "    In order to graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-2y%3E=4 ALIGN=MIDDLE  ALT=\"x-2y%3E=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > we need to graph the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-2y=4 ALIGN=MIDDLE  ALT=\"x-2y=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > (just replace the inequality sign with an equal sign). So lets graph the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-2y=4 ALIGN=MIDDLE  ALT=\"x-2y=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > (note: if you need help with graphing, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Linear-equations/graphing-linear-equations.solver>solver<a>)\r<BR>\n" );
document.write( "\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%284-1%2Ax%29%2F-2%29 ALIGN=MIDDLE  ALT=\"graph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%284-1%2Ax%29%2F-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-2y=4 ALIGN=MIDDLE  ALT=\"x-2y=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "    Now lets pick a test point, say (0,0) (any point will work, but this point is the easiest to work with), and evaluate the inequality <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-2y%3E=4 ALIGN=MIDDLE  ALT=\"x-2y%3E=4\"   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=%280%29-2%280%29%3E=4 ALIGN=MIDDLE  ALT=\"%280%29-2%280%29%3E=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in x=0, y=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "     <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3E=4 ALIGN=MIDDLE  ALT=\"0%3E=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since this inequality is <b>not</b> true, we shade the entire region that <b>doesn't</b> contain (0,0)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2Cgraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%284-1%2Ax%29%2F-2%2C+%284-1%2Ax%29%2F-2-0%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%284-1%2Ax%29%2F-2%2C+%284-1%2Ax%29%2F-2-0.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%284-1%2Ax%29%2F-2%2C+%284-1%2Ax%29%2F-2-1.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%284-1%2Ax%29%2F-2%2C+%284-1%2Ax%29%2F-2-2.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%284-1%2Ax%29%2F-2%2C+%284-1%2Ax%29%2F-2-3%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%284-1%2Ax%29%2F-2%2C+%284-1%2Ax%29%2F-2-3.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%284-1%2Ax%29%2F-2%2C+%284-1%2Ax%29%2F-2-4.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%284-1%2Ax%29%2F-2%2C+%284-1%2Ax%29%2F-2-5.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%284-1%2Ax%29%2F-2%2C+%284-1%2Ax%29%2F-2-6%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%284-1%2Ax%29%2F-2%2C+%284-1%2Ax%29%2F-2-6.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%284-1%2Ax%29%2F-2%2C+%284-1%2Ax%29%2F-2-7.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%284-1%2Ax%29%2F-2%2C+%284-1%2Ax%29%2F-2-8.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%284-1%2Ax%29%2F-2%2C+%284-1%2Ax%29%2F-2-9%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%284-1%2Ax%29%2F-2%2C+%284-1%2Ax%29%2F-2-9.75%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( " Here is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-2y%3E=4 ALIGN=MIDDLE  ALT=\"x-2y%3E=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > with the graph of the line(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-2y=4 ALIGN=MIDDLE  ALT=\"x-2y=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >) in red and the shaded region in green\r<BR>\n" );
document.write( "\n" );
document.write( "(note: In this case, the red line is a solid line. This means the boundaries are included in the region.)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now lets graph the second inequality\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "    In order to graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C=4 ALIGN=MIDDLE  ALT=\"x%3C=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > we need to graph the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > (just replace the inequality sign with an equal sign). So lets graph the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > (note: if you need help with graphing, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Linear-equations/graphing-linear-equations.solver>solver<a>)\r<BR>\n" );
document.write( "\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C1000%28x-4000%2F1000%29%29 ALIGN=MIDDLE  ALT=\"graph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C1000%28x-4000%2F1000%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "    Now lets pick a test point, say (0,0) (any point will work, but this point is the easiest to work with), and evaluate the inequality <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C=4 ALIGN=MIDDLE  ALT=\"x%3C=4\"   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=%280%29%3C=4 ALIGN=MIDDLE  ALT=\"%280%29%3C=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in x=0, y=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "     <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C=4 ALIGN=MIDDLE  ALT=\"0%3C=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since this inequality is true, we shade the entire region containing (0,0)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2Cgraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-4000%2F1000%29%2C+10000%28x-4000%2F1000%2B0%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-4000%2F1000%29%2C+10000%28x-4000%2F1000%2B0.75%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-4000%2F1000%29%2C+10000%28x-4000%2F1000%2B1.5%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-4000%2F1000%29%2C+10000%28x-4000%2F1000%2B2.25%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-4000%2F1000%29%2C+10000%28x-4000%2F1000%2B3%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-4000%2F1000%29%2C+10000%28x-4000%2F1000%2B3.75%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-4000%2F1000%29%2C+10000%28x-4000%2F1000%2B4.5%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-4000%2F1000%29%2C+10000%28x-4000%2F1000%2B5.25%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-4000%2F1000%29%2C+10000%28x-4000%2F1000%2B6%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-4000%2F1000%29%2C+10000%28x-4000%2F1000%2B6.75%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-4000%2F1000%29%2C+10000%28x-4000%2F1000%2B7.5%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-4000%2F1000%29%2C+10000%28x-4000%2F1000%2B8.25%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-4000%2F1000%29%2C+10000%28x-4000%2F1000%2B9%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-4000%2F1000%29%2C+10000%28x-4000%2F1000%2B9.75%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( " Here is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C=4 ALIGN=MIDDLE  ALT=\"x%3C=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > with the graph of the line(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >) in red and the shaded region in green\r<BR>\n" );
document.write( "\n" );
document.write( "(note: In this case, the red line is a solid line. This means the boundaries are included in the region.)\r<BR>\n" );
document.write( "\n" );
document.write( "So we essentially have these 2 regions\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2Cgraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%284-1%2Ax%29%2F-2%2C+%284-1%2Ax%29%2F-2-0%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%284-1%2Ax%29%2F-2%2C+%284-1%2Ax%29%2F-2-0.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%284-1%2Ax%29%2F-2%2C+%284-1%2Ax%29%2F-2-1.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%284-1%2Ax%29%2F-2%2C+%284-1%2Ax%29%2F-2-2.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%284-1%2Ax%29%2F-2%2C+%284-1%2Ax%29%2F-2-3%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%284-1%2Ax%29%2F-2%2C+%284-1%2Ax%29%2F-2-3.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%284-1%2Ax%29%2F-2%2C+%284-1%2Ax%29%2F-2-4.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%284-1%2Ax%29%2F-2%2C+%284-1%2Ax%29%2F-2-5.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%284-1%2Ax%29%2F-2%2C+%284-1%2Ax%29%2F-2-6%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%284-1%2Ax%29%2F-2%2C+%284-1%2Ax%29%2F-2-6.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%284-1%2Ax%29%2F-2%2C+%284-1%2Ax%29%2F-2-7.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%284-1%2Ax%29%2F-2%2C+%284-1%2Ax%29%2F-2-8.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%284-1%2Ax%29%2F-2%2C+%284-1%2Ax%29%2F-2-9%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%284-1%2Ax%29%2F-2%2C+%284-1%2Ax%29%2F-2-9.75%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Region #1 which is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-2y%3E=4 ALIGN=MIDDLE  ALT=\"x-2y%3E=4\"   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=drawing%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2Cgraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-4000%2F1000%29%2C+10000%28x-4000%2F1000%2B0%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-4000%2F1000%29%2C+10000%28x-4000%2F1000%2B0.75%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-4000%2F1000%29%2C+10000%28x-4000%2F1000%2B1.5%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-4000%2F1000%29%2C+10000%28x-4000%2F1000%2B2.25%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-4000%2F1000%29%2C+10000%28x-4000%2F1000%2B3%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-4000%2F1000%29%2C+10000%28x-4000%2F1000%2B3.75%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-4000%2F1000%29%2C+10000%28x-4000%2F1000%2B4.5%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-4000%2F1000%29%2C+10000%28x-4000%2F1000%2B5.25%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-4000%2F1000%29%2C+10000%28x-4000%2F1000%2B6%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-4000%2F1000%29%2C+10000%28x-4000%2F1000%2B6.75%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-4000%2F1000%29%2C+10000%28x-4000%2F1000%2B7.5%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-4000%2F1000%29%2C+10000%28x-4000%2F1000%2B8.25%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-4000%2F1000%29%2C+10000%28x-4000%2F1000%2B9%29%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C10000%28x-4000%2F1000%29%2C+10000%28x-4000%2F1000%2B9.75%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Region #2 which is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C=4 ALIGN=MIDDLE  ALT=\"x%3C=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So these regions overlap to produce this region. It's a little hard to see, but after evenly shading each region, the intersecting region will be the most shaded in.\r<BR>\n" );
document.write( "\n" );
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document.write( "\n" );
document.write( "Here is a cleaner look at the intersection of regions\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
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ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Here is the intersection of the 2 regions represented by the dots </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );