document.write( "Question 85712: Solve the following system of linear inequalities by graphing.
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\n" ); document.write( " \"x+%2B+2y+%3E=+6\"
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Algebra.Com's Answer #61910 by jim_thompson5910(35256)\"\" \"About 
You can put this solution on YOUR website!
11.\r
\n" ); document.write( "\n" ); document.write( "Start with the given system of inequalities\r
\n" ); document.write( "\n" ); document.write( "\"x-y%3C=3\"\r
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\n" ); document.write( "\n" ); document.write( "\"x%2B2y%3E=6\"\r
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\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
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\n" ); document.write( "\n" ); document.write( "So lets graph the first inequality\r
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\n" ); document.write( "\n" ); document.write( " In order to graph \"x-y%3C=3\" we need to graph the equation \"x-y=3\" (just replace the inequality sign with an equal sign). So lets graph the line \"x-y=3\" (note: if you need help with graphing, check out this solver)\r
\n" ); document.write( "\n" ); document.write( " \"graph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F-1%29\" graph of \"x-y=3\"\r
\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 \"x-y%3C=3\"\r
\n" ); document.write( "\n" ); document.write( " \"%280%29-%280%29%3C=3\" Plug in x=0, y=0\r
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\n" ); document.write( "\n" ); document.write( " \"0%3C=3\" Simplify\r
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\n" ); document.write( "\n" ); document.write( "Since this inequality is true, we shade the entire region containing (0,0)\r
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\n" ); document.write( "\n" ); document.write( " Here is the graph of \"x-y%3C=3\" with the graph of the line(\"x-y=3\") in red and the shaded region in green\r
\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
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\n" ); document.write( "\n" ); document.write( "Now lets graph the second inequality\r
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\n" ); document.write( "\n" ); document.write( " In order to graph \"x%2B2y%3E=6\" we need to graph the equation \"x%2B2y=6\" (just replace the inequality sign with an equal sign). So lets graph the line \"x%2B2y=6\" (note: if you need help with graphing, check out this
solver)\r
\n" ); document.write( "\n" ); document.write( " \"graph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-1%2Ax%29%2F2%29\" graph of \"x%2B2y=6\"\r
\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 \"x%2B2y%3E=6\"\r
\n" ); document.write( "\n" ); document.write( " \"%280%29%2B2%280%29%3E=6\" Plug in x=0, y=0\r
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\n" ); document.write( "\n" ); document.write( " \"0%3E=6\" Simplify\r
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\n" ); document.write( "\n" ); document.write( "Since this inequality is not true, we shade the entire region that doesn't contain (0,0)\r
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\n" ); document.write( "\n" ); document.write( " Here is the graph of \"x%2B2y%3E=6\" with the graph of the line(\"x%2B2y=6\") in red and the shaded region in green\r
\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
\n" ); document.write( "\n" ); document.write( "So we essentially have these 2 regions\r
\n" ); document.write( "\n" ); document.write( " Region #1 which is the graph of \"x-y%3C=3\"\r
\n" ); document.write( "\n" ); document.write( " Region #2 which is the graph of \"x%2B2y%3E=6\"\r
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\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
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\n" ); document.write( "\n" ); document.write( "Here is a cleaner look at the intersection of regions\r
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\n" ); document.write( "\n" ); document.write( " Here is the intersection of the 2 regions represented by the dots
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