document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=169654>Question 169654</A>: <I><SPAN STYLE=\"word-break: break-all;\"> <a  rel=nofollow HREF=\"http://s165.photobucket.com/albums/u56/regularkyle/?action=view&current=mathquestioon.jpg\" target=\"_blank\"><IMG STYLE=\"max-width:80%;\" src=\"http://i165.photobucket.com/albums/u56/regularkyle/mathquestioon.jpg\" border=\"0\" alt=\"Photobucket\"></a>\r<BR>\n" );
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document.write( "my question is :\r<BR>\n" );
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document.write( "a student graphed a system of inequalities, shading only the region containong the solutions of the sytem. Write a system of inequalities corresponding to the students graph. Explain how you arrived at your answer. \r<BR>\n" );
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document.write( "there is no isbn because this is a module book question </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #125026 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=125026\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There are two lines graphed, so we need to find the equations of those two lines first. From there, we can find the inequalities.\r<BR>\n" );
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document.write( "Line 1: Line going through points (0,4) and (4,0)\r<BR>\n" );
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document.write( "So let's find the equation of line 1\r<BR>\n" );
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document.write( "First let's find the slope of the line through the points <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(0,4\right)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(4,0\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 ALIGN=MIDDLE  ALT=\"m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the slope formula.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=%280-4%29%2F%284-0%29 ALIGN=MIDDLE  ALT=\"m=%280-4%29%2F%284-0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B2%5D=0 ALIGN=MIDDLE  ALT=\"y%5B2%5D=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D=4 ALIGN=MIDDLE  ALT=\"y%5B1%5D=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D=4 ALIGN=MIDDLE  ALT=\"x%5B2%5D=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D=0 ALIGN=MIDDLE  ALT=\"x%5B1%5D=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=m=%28-4%29%2F%284-0%29 ALIGN=MIDDLE  ALT=\"m=%28-4%29%2F%284-0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   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=m=%28-4%29%2F%284%29 ALIGN=MIDDLE  ALT=\"m=%28-4%29%2F%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   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=m=-1 ALIGN=MIDDLE  ALT=\"m=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce\r<BR>\n" );
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document.write( "So the slope of the line that goes through the points <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(0,4\right)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(4,0\right)\"> is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=-1 ALIGN=MIDDLE  ALT=\"m=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now let's use the point slope formula:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-y%5B1%5D=m%28x-x%5B1%5D%29 ALIGN=MIDDLE  ALT=\"y-y%5B1%5D=m%28x-x%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the point slope formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-4=-1%28x-0%29 ALIGN=MIDDLE  ALT=\"y-4=-1%28x-0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=-1 ALIGN=MIDDLE  ALT=\"m=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D=0 ALIGN=MIDDLE  ALT=\"x%5B1%5D=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D=4 ALIGN=MIDDLE  ALT=\"y%5B1%5D=4\"   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-4=-1x%2B-1%28-0%29 ALIGN=MIDDLE  ALT=\"y-4=-1x%2B-1%28-0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-4=-1x%2B0 ALIGN=MIDDLE  ALT=\"y-4=-1x%2B0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-1x%2B0%2B4 ALIGN=MIDDLE  ALT=\"y=-1x%2B0%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add 4 to both sides. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-1x%2B4 ALIGN=MIDDLE  ALT=\"y=-1x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms. \r<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;\" > Simplify\r<BR>\n" );
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document.write( "So the equation that goes through the points <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(0,4\right)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(4,0\right)\"> is <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;\" >\r<BR>\n" );
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document.write( "-------------------------------------------------------\r<BR>\n" );
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document.write( "Line 2: Line through the points (0,0) and (3,6)\r<BR>\n" );
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document.write( "So let's find the equation of line 2\r<BR>\n" );
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document.write( "First let's find the slope of the line through the points <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(0,0\right)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(3,6\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 ALIGN=MIDDLE  ALT=\"m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the slope formula.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=%286-0%29%2F%283-0%29 ALIGN=MIDDLE  ALT=\"m=%286-0%29%2F%283-0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B2%5D=6 ALIGN=MIDDLE  ALT=\"y%5B2%5D=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D=0 ALIGN=MIDDLE  ALT=\"y%5B1%5D=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D=3 ALIGN=MIDDLE  ALT=\"x%5B2%5D=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D=0 ALIGN=MIDDLE  ALT=\"x%5B1%5D=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=m=%286%29%2F%283-0%29 ALIGN=MIDDLE  ALT=\"m=%286%29%2F%283-0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"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=m=%286%29%2F%283%29 ALIGN=MIDDLE  ALT=\"m=%286%29%2F%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > from <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 get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"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=m=2 ALIGN=MIDDLE  ALT=\"m=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce\r<BR>\n" );
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document.write( "So the slope of the line that goes through the points <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(0,0\right)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(3,6\right)\"> is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=2 ALIGN=MIDDLE  ALT=\"m=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now let's use the point slope formula:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-y%5B1%5D=m%28x-x%5B1%5D%29 ALIGN=MIDDLE  ALT=\"y-y%5B1%5D=m%28x-x%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the point slope formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-0=2%28x-0%29 ALIGN=MIDDLE  ALT=\"y-0=2%28x-0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=2 ALIGN=MIDDLE  ALT=\"m=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D=0 ALIGN=MIDDLE  ALT=\"x%5B1%5D=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D=0 ALIGN=MIDDLE  ALT=\"y%5B1%5D=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-0=2x%2B2%28-0%29 ALIGN=MIDDLE  ALT=\"y-0=2x%2B2%28-0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-0=2x%2B0 ALIGN=MIDDLE  ALT=\"y-0=2x%2B0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x%2B0%2B0 ALIGN=MIDDLE  ALT=\"y=2x%2B0%2B0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add 0 to both sides. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x%2B0 ALIGN=MIDDLE  ALT=\"y=2x%2B0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x ALIGN=MIDDLE  ALT=\"y=2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Remove the trailing zero\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x ALIGN=MIDDLE  ALT=\"y=2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify\r<BR>\n" );
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document.write( "So the equation that goes through the points <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(0,0\right)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(3,6\right)\"> is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x ALIGN=MIDDLE  ALT=\"y=2x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the system of <b>equations</b>(we'll convert them into inequalities later) is:\r<BR>\n" );
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document.write( "Now, notice how the shaded region is BELOW both of the lines. In other words, EVERY point in the shaded region has a y-coordinate that is less than a point on either line.\r<BR>\n" );
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document.write( "So this means that we can take <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;\" > and convert it to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3C-x%2B4 ALIGN=MIDDLE  ALT=\"y%3C-x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since every point in this shaded region is below the line <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;\" >) and we can take <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x ALIGN=MIDDLE  ALT=\"y=2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and convert it to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3C2x ALIGN=MIDDLE  ALT=\"y%3C2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since every point in this shaded region is below the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x ALIGN=MIDDLE  ALT=\"y=2x\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "So we then end up with the system of inequalities\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y%3C-x%2B4%2Cy%3C2x%29 ALIGN=MIDDLE  ALT=\"system%28y%3C-x%2B4%2Cy%3C2x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Note: If you are INCLUDING the boundaries, then you need to change the inequality signs from <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE <\">  to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \le\"> like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y%3C=-x%2B4%2Cy%3C=2x%29 ALIGN=MIDDLE  ALT=\"system%28y%3C=-x%2B4%2Cy%3C=2x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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