\r\n" );
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document.write( "Graph the absolute value inequalities:\r\n" );
document.write( "|x + 2y| > 6 \r\n" );
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document.write( "The boundary equation is found by replacing\r\n" );
document.write( "the inequality symbol by an equal sign:\r\n" );
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document.write( "|x + 2y| = 6\r\n" );
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document.write( "which is equivalent to\r\n" );
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document.write( "x + 2y = 6 OR x + 2y = -6\r\n" );
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document.write( "So we graph those two boundary lines dotted\r\n" );
document.write( "since the inequality is > and not >.\r\n" );
document.write( "This says that the boundary lines are not\r\n" );
document.write( "included in the graph.\r\n" );
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\r\n" );
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document.write( "The boundary lines divide the xy-system into \r\n" );
document.write( "three regions:\r\n" );
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document.write( "1. the region above the upper dotted line.\r\n" );
document.write( "2. the region between the two dotted lines.\r\n" );
document.write( "3. the region below the lower dotted line.\r\n" );
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document.write( "Now we find out which of these regions \r\n" );
document.write( "contain the solutions and which ones don't,\r\n" );
document.write( "so that we can shade the regions that do, \r\n" );
document.write( "and leave the regions that don't unshaded.\r\n" );
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document.write( "So we pick an arbitrary test point in each of \r\n" );
document.write( "the three regions and substitute it into the \r\n" );
document.write( "original inequality. If we get a true inequality,\r\n" );
document.write( "then we shade that region. If we get a false \r\n" );
document.write( "inequality, then we do not shade that region.\r\n" );
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document.write( "In the upper region, let's pick as a test point, \r\n" );
document.write( "say, the point (6,5), and substitute x=6 and y=5 \r\n" );
document.write( "into the original inequality:\r\n" );
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document.write( " |x + 2y| > 6\r\n" );
document.write( "|6 + 2(5)| > 6\r\n" );
document.write( " |6 + 10| > 6\r\n" );
document.write( " |16| > 6\r\n" );
document.write( " 16 > 6\r\n" );
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document.write( "This is true so the test point (6,5) is a solution,\r\n" );
document.write( "so we shade the upper region.\r\n" );
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document.write( "In the middle region, that is, the region between \r\n" );
document.write( "the two lines, let's pick as a test point, say, \r\n" );
document.write( "the point (1,1), and substitute x=1 and y=1 into \r\n" );
document.write( "the original inequality:\r\n" );
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document.write( " |x + 2y| > 6\r\n" );
document.write( "|1 + 2(1)| > 6\r\n" );
document.write( " |1 + 2| > 6\r\n" );
document.write( " |3| > 6\r\n" );
document.write( " 3 > 6\r\n" );
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document.write( "That is false so this test point (1,1) is not a \r\n" );
document.write( "solution so we do not shade the region between \r\n" );
document.write( "the two lines.\r\n" );
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document.write( "In the lower region, let's pick as a test point, say,\r\n" );
document.write( "the point (3,-7), and substitute x=3 and y=-7 into \r\n" );
document.write( "the original inequality:\r\n" );
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document.write( " |x + 2y| > 6\r\n" );
document.write( "|3 + 2(-7)| > 6\r\n" );
document.write( " |3 - 14| > 6\r\n" );
document.write( " |-11| > 6\r\n" );
document.write( " 11 > 6\r\n" );
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document.write( "This is true so, the test point (3,-7) is a solution\r\n" );
document.write( "so we shade the lower region.\r\n" );
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document.write( "I can't shade on here, but you can on your paper:\r\n" );
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document.write( "
\r\n" );
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document.write( "Edwin
\n" );
document.write( "![\"\"](\"/images/stars/stars-13.gif\")
