document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=724251>Question 724251</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Shade the solution set to the system of inequalities<BR>\n" );
document.write( " x + 3y > -2<BR>\n" );
document.write( "2x + y less than or equal to 2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #443708 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=443708\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28+x%2B+3y+%3E+-2%2C2x+%2B+y+%3C=2%29 ALIGN=MIDDLE  ALT=\"system%28+x%2B+3y+%3E+-2%2C2x+%2B+y+%3C=2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We need to graph the line for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B+3y=-2 ALIGN=MIDDLE  ALT=\"x%2B+3y=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > as a dashed line to show that the points on that line are not part of the solution.<BR>\n" );
document.write( "We need to graph part of the line for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x+%2B+y+=2 ALIGN=MIDDLE  ALT=\"2x+%2B+y+=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > as a solid line to show that those points are part of the solution.<BR>\n" );
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document.write( "We need two points to determine each line, and I will find points that are easy to calculate and give me a good graph.<BR>\n" );
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document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B+3y=-2 ALIGN=MIDDLE  ALT=\"x%2B+3y=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "<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;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2 ALIGN=MIDDLE  ALT=\"x=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives us point (-2,0)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2 ALIGN=MIDDLE  ALT=\"y=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-6=-2 ALIGN=MIDDLE  ALT=\"x-6=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2%2B6 ALIGN=MIDDLE  ALT=\"x=-2%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <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;\" > gives us point (4,-2)<BR>\n" );
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document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x+%2B+y+=2 ALIGN=MIDDLE  ALT=\"2x+%2B+y+=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "<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;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2 ALIGN=MIDDLE  ALT=\"y=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives us point (0,2)<BR>\n" );
document.write( "<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;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x=2 ALIGN=MIDDLE  ALT=\"2x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives us point (1,0)<BR>\n" );
document.write( "Those two points are a bit too close together, but may still give us a decent graph.<BR>\n" );
document.write( "Seeing the lines plotted may make what follows easier.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-5%2C5%2C-5%2C5%2C%0D%0Agrid%281%29%2C%0D%0Ablue%28circle%28-2%2C0%2C0.2%29%29%2Cblue%28circle%284%2C-2%2C0.2%29%29%2C%0D%0Ablue%28line%28-8%2C2%2C10%2C-4%29%29%2C%0D%0Agreen%28circle%280%2C2%2C0.2%29%29%2Cgreen%28circle%281%2C0%2C0.2%29%29%2C%0D%0Agreen%28line%28-5%2C12%2C6%2C-10%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > The solution will include all the points inside one of the four angles formed by the lines.<BR>\n" );
document.write( "Using a test point that is not part of the lines would allow us to see if that point is on the same side of each line as the solution.<BR>\n" );
document.write( "The easiest test point is (0,0), the origin, with <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;\" ><BR>\n" );
document.write( "Substituting, we find that (0,0) is part of the solution for both inequalities:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B3y=0%2B3%2A0=0%3E-2 ALIGN=MIDDLE  ALT=\"x%2B3y=0%2B3%2A0=0%3E-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2By=2%2A0%2B0=0%3C2 ALIGN=MIDDLE  ALT=\"2x%2By=2%2A0%2B0=0%3C2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So we shade the wedge that contains the origin, and plot the lines, dashed and solid as needed.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-5%2C5%2C-5%2C5%2C%0D%0Agrid%281%29%2C%0D%0Ablue%28line%28-5%2C1%2C-3.8%2C0.6%29%29%2Cblue%28line%28-2.6%2C0.2%2C-1.4%2C-0.2%29%29%2C%0D%0Ablue%28line%28-0.2%2C-0.6%2C1%2C-1%29%29%2Cblue%28line%282.2%2C-1.4%2C3.4%2C-1.8%29%29%2C%0D%0Agreen%28circle%281.6%2C-1.2%2C0.2%29%29%2Cblue%28line%284.6%2C-2.2%2C5.8%2C-2.6%29%29%2C%0D%0Agreen%28line%28-1.5%2C5%2C1.5%2C-1%29%29%2C%0D%0Agreen%28line%281.9%2C-1.8%2C2.3%2C-2.6%29%29%2Cgreen%28line%282.9%2C-3.8%2C3.3%2C-4.6%29%29%2C%0D%0Agreen%28line%281%2C-1%2C1.5%2C-1%29%29%2Cgreen%28line%28-0.2%2C-0.6%2C1.3%2C-0.6%29%29%2C%0D%0Agreen%28line%28-1.4%2C-0.2%2C1.1%2C-0.2%29%29%2Cgreen%28line%28-2.6%2C0.2%2C0.9%2C0.2%29%29%2C%0D%0Agreen%28line%28-3.8%2C0.6%2C0.7%2C0.6%29%29%2Cgreen%28line%28-5%2C1%2C0.5%2C1%29%29%2C%0D%0Agreen%28line%28-5%2C1.4%2C0.3%2C1.4%29%29%2Cgreen%28line%28-5%2C1.8%2C0.1%2C1.8%29%29%2C%0D%0Agreen%28line%28-5%2C2.2%2C-0.1%2C2.2%29%29%2Cgreen%28line%28-5%2C2.6%2C-0.3%2C2.6%29%29%2C%0D%0Agreen%28line%28-5%2C3%2C-0.5%2C3%29%29%2Cgreen%28line%28-5%2C3.4%2C-0.7%2C3.4%29%29%2C%0D%0Agreen%28line%28-5%2C3.8%2C-0.9%2C3.8%29%29%2Cgreen%28line%28-5%2C4.2%2C-1.1%2C4.2%29%29%2C%0D%0Agreen%28line%28-5%2C4.6%2C-1.3%2C4.6%29%29%2Cgreen%28line%28-5%2C5%2C-1.5%2C5%29%29%2C%0D%0Agreen%28line%280.4%2C-0.8%2C1.4%2C-0.8%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > The intersection of the two lines is not part of the solution.<BR>\n" );
document.write( "Its coordinates satisfy the equations of both lines, including <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B+3y=-2 ALIGN=MIDDLE  ALT=\"x%2B+3y=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, but it do not satisfy the inequality <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B+3y%3E-2 ALIGN=MIDDLE  ALT=\"x%2B+3y%3E-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );