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document.write( "Graph the inequality.\r\n" );
document.write( "3x + y < 5\r\n" );
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document.write( "First form the equation of the boundary line,\r\n" );
document.write( "which is found by replacing the < by =\r\n" );
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document.write( " 3x + y = 5\r\n" );
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document.write( "Draw this line by find the intercept (0,5) and (3,-4)\r\n" );
document.write( "Draw the line solid, not dotted, because the inequality is\r\n" );
document.write( "< and not <.\r\n" );
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\r\n" );
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document.write( "Now we only have to decide whether we are to shade the \r\n" );
document.write( "right side of the line or the left side of it. To do\r\n" );
document.write( "this we pick any point which IS NOT on the line. Say we\r\n" );
document.write( "choose as a test point the point (2,3) marked below with \r\n" );
document.write( "an o, which is on the right side of the line.\r\n" );
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document.write( "
\r\n" );
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document.write( "Now we substitute \r\n" );
document.write( "(x,y) = (2,3) into the inequality\r\n" );
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document.write( " 3x + y < 5\r\n" );
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document.write( "If we get a true inequality, then we will shade the\r\n" );
document.write( "same side of the line that the test point is on. If\r\n" );
document.write( "we get a false inequality, we shade the other side:\r\n" );
document.write( "\r\n" );
document.write( " 3x + y < 5\r\n" );
document.write( " 3(2) + (3) < 5 \r\n" );
document.write( " 6 + 3 < 5\r\n" );
document.write( " 9 < 5\r\n" );
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document.write( "This is false, so we shade the side OPPOSITE the\r\n" );
document.write( "side of the line the test point that we picked is on,\r\n" );
document.write( "namely the left side. I can't shade on here but you can on\r\n" );
document.write( "your paper: \r\n" );
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\r\n" );
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document.write( "A shortcut would have been to have chosen the origin\r\n" );
document.write( "(0,0) as the test point. You could have. Then you\r\n" );
document.write( "could have substituted it in your head to decide\r\n" );
document.write( "which side of the line to shade. Substituting\r\n" );
document.write( "(x,y) = (0,0) you would have gotten simply\r\n" );
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document.write( " 3x + y < 5\r\n" );
document.write( " 3(0) + (0) < 5 \r\n" );
document.write( " 0 + 0 < 5\r\n" );
document.write( " 0 < 5\r\n" );
document.write( " \r\n" );
document.write( "which is TRUE and which you could have done in your\r\n" );
document.write( "head! Since it is TRUE, you would have known to\r\n" );
document.write( "shade the side of the line which the origin (0,0)\r\n" );
document.write( "is on. Notice that the origin (0,0) is on the left side\r\n" );
document.write( "of the line, and that is exactly the same side we \r\n" );
document.write( "determined that we should shade by using the test point\r\n" );
document.write( "(2,3). \r\n" );
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document.write( "So why didn't I choose the origin (0,0) as a test point?\r\n" );
document.write( "I would have normally, if I weren't teaching you. But\r\n" );
document.write( "if I had you might have gotten the wrong idea that you can \r\n" );
document.write( "ALWAYS choose the origin as a test point. But you cannot \r\n" );
document.write( "choose the origin as a test point when the line passes \r\n" );
document.write( "through the origin. However that is the only \r\n" );
document.write( "time you cannot choose the origin (0,0) as a test point.\r\n" );
document.write( "So feel free to use the origin as a test point except\r\n" );
document.write( "when the line goes through the origin.\r\n" );
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document.write( "Edwin
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document.write( "![\"\"](\"/images/stars/stars-13.gif\")
