document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=151526>Question 151526</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the system of inequalities by graphing the solution region:\r<BR>\n" );
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document.write( "{  x+2y<_ 3<BR>\n" );
document.write( "  2x-3y<_6   } </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #111395 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=111395\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
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document.write( "<a  rel=nofollow HREF=\"#1\">Jump to Inequality #1</a><BR>\n" );
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document.write( "<a  rel=nofollow HREF=\"#solution\">Jump to Solution</a>\r<BR>\n" );
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document.write( "Inequality #1 \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;\" > Start with the 1st inequality.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2y%3C=3-x ALIGN=MIDDLE  ALT=\"2y%3C=3-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.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3C=%283-x%29%2F%282%29 ALIGN=MIDDLE  ALT=\"y%3C=%283-x%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to isolate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   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%3C=-%281%2F2%29x%2B3%2F2 ALIGN=MIDDLE  ALT=\"y%3C=-%281%2F2%29x%2B3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify.\r<BR>\n" );
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document.write( "So in order to plot <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;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3C=-%281%2F2%29x%2B3%2F2 ALIGN=MIDDLE  ALT=\"y%3C=-%281%2F2%29x%2B3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (which is the same thing), we need to plot the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%281%2F2%29x%2B3%2F2 ALIGN=MIDDLE  ALT=\"y=-%281%2F2%29x%2B3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > first. <BR>\n" );
document.write( "(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+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%29+ ALIGN=MIDDLE  ALT=\"+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%281%2F2%29x%2B3%2F2 ALIGN=MIDDLE  ALT=\"y=-%281%2F2%29x%2B3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now plug in a test point (0,0) into the inequality <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3C=-%281%2F2%29x%2B3%2F2 ALIGN=MIDDLE  ALT=\"y%3C=-%281%2F2%29x%2B3%2F2\"   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%3C=-%281%2F2%29x%2B3%2F2 ALIGN=MIDDLE  ALT=\"y%3C=-%281%2F2%29x%2B3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given inequality.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C=-%281%2F2%29%2A%280%29%2B3%2F2 ALIGN=MIDDLE  ALT=\"0%3C=-%281%2F2%29%2A%280%29%2B3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <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;\" > and <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;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C=3%2F2 ALIGN=MIDDLE  ALT=\"0%3C=3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Evaluate and simplify.\r<BR>\n" );
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document.write( "Since the inequality is true, this means that we shade the entire region that contains the point (0,0)\r<BR>\n" );
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document.write( "In other words, we simply shade the <b>entire</b> region that is below the line.\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%0D%0A++++graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-0%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-1%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-2%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-3%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-4%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-5%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-6%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-7%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-8%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-9%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-10%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-11%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-12%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-13%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-14%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-15%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-16%29%0D%0A%0D%0A+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3C=-%281%2F2%29x%2B3%2F2 ALIGN=MIDDLE  ALT=\"y%3C=-%281%2F2%29x%2B3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > with the shaded region in green\r<BR>\n" );
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document.write( "Now move onto the next inequality\r<BR>\n" );
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document.write( "Inequality #2 \r<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;\" > Start with the 2nd inequality.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3y%3C=6-2x ALIGN=MIDDLE  ALT=\"-3y%3C=6-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x ALIGN=MIDDLE  ALT=\"2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3E=%286-2x%29%2F%28-3%29 ALIGN=MIDDLE  ALT=\"y%3E=%286-2x%29%2F%28-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to isolate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This will flip the inequality sign.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3E=%282%2F3%29x-2 ALIGN=MIDDLE  ALT=\"y%3E=%282%2F3%29x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify.\r<BR>\n" );
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document.write( "So in order to plot <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;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3E=%282%2F3%29x-2 ALIGN=MIDDLE  ALT=\"y%3E=%282%2F3%29x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (which is the same thing), we need to plot the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2F3%29x-2 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > first. <BR>\n" );
document.write( "(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+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%29+ ALIGN=MIDDLE  ALT=\"+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2F3%29x-2 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now plug in a test point (0,0) into the inequality <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3E=%282%2F3%29x-2 ALIGN=MIDDLE  ALT=\"y%3E=%282%2F3%29x-2\"   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=%282%2F3%29x-2 ALIGN=MIDDLE  ALT=\"y%3E=%282%2F3%29x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given inequality.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3E=%282%2F3%29%2A%280%29-2 ALIGN=MIDDLE  ALT=\"0%3E=%282%2F3%29%2A%280%29-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <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;\" > and <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;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3E=-2 ALIGN=MIDDLE  ALT=\"0%3E=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Evaluate and simplify.\r<BR>\n" );
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document.write( "Since the inequality is true, this means that we shade the entire region that contains the point (0,0)\r<BR>\n" );
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document.write( "In other words, we simply shade the <b>entire</b> region that is above the line.\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%0D%0A++++graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B0%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B1.33333333333333%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B2.66666666666667%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B4%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B5.33333333333333%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B6.66666666666667%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B8%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B9.33333333333333%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B10.6666666666667%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B12%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B13.3333333333333%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B14.6666666666667%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B16%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B17.3333333333333%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B18.6666666666667%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B20%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B21.3333333333333%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B22.6666666666667%29%0D%0A%0D%0A+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3E=%282%2F3%29x-2 ALIGN=MIDDLE  ALT=\"y%3E=%282%2F3%29x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > with the shaded region in green\r<BR>\n" );
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document.write( " Now draw all of the inequalities on the same coordinate system.\r<BR>\n" );
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document.write( "  The solution region will be the intersecting region of the inequalities.\r<BR>\n" );
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document.write( "  ----------------------------  <font size=\"4\" color=\"red\"><b>Solution</b></font>  --------------------------\r<BR>\n" );
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document.write( " So the solution looks like:\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" 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