document.write( "Question 80392: Solve each of the following systems of linear inequalities graphically.\r
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Algebra.Com's Answer #57683 by Edwin McCravy(20056) $\"\"$ $\"About$
You can put this solution on YOUR website!
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document.write( "Solve each of the following systems of linear \r\n" );
document.write( "inequalities graphically. \r\n" );
document.write( "10.\r\n" );
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document.write( "3x – y < 6\r\n" );
document.write( "x > 1\r\n" );
document.write( "y < 3\r\n" );
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document.write( "First form the three equations of the boundary lines.\r\n" );
document.write( "These are formed by replacing each inequality symbol\r\n" );
document.write( "by an equal sign. The equations of the boundary lines\r\n" );
document.write( "are \r\n" );
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document.write( "3x – y = 6\r\n" );
document.write( "     x = 1\r\n" );
document.write( "     y = 3\r\n" );
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document.write( "Now we draw the equation of the first boundary line,\r\n" );
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document.write( "3x - y = 6. It's intercepts are (0,-6) and (2,0):\r\n" );
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document.write( "It should be a dotted line, not a solid line.\r\n" );
document.write( "I cannot draw dotted lines on here, but you can on\r\n" );
document.write( "your paper.\r\n" );
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document.write( "But now before drawing the second line we must decide\r\n" );
document.write( "which side of this boudary line the solutions to the\r\n" );
document.write( "corresponding inequality the solutions are on. To do\r\n" );
document.write( "this we arbitrarily select any point that DOES NOT lie \r\n" );
document.write( "on the boundary line as a test point. \r\n" );
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document.write( "Suppose we choose the point (-3,5) as a test point.\r\n" );
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document.write( "Now we substitute (-3,5) into the inequality\r\n" );
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document.write( "   3x – y < 6\r\n" );
document.write( "3(-2) - 5 < 6\r\n" );
document.write( "   -6 - 5 < 6\r\n" );
document.write( "      -11 < 6\r\n" );
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document.write( "This is true, so its solutions lie on the\r\n" );
document.write( "same side of the boundary line that the test \r\n" );
document.write( "poin was on.  Therefore we make some marks\r\n" );
document.write( "along that side of the line:\r\n" );
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document.write( "Now we're ready to draw the next boundary\r\n" );
document.write( "line, x = 1. This is a vertical line that\r\n" );
document.write( "intersects the x axis at 1. Again it\r\n" );
document.write( "should be a dotted line, and you should\r\n" );
document.write( "draw it dotted on your paper:\r\n" );
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document.write( "We can use the same test point we used for the\r\n" );
document.write( "first line since it is not on the line.\r\n" );
document.write( "So we substitute (-3,5) into the inequality.\r\n" );
document.write( "Not that we need only substitute the x-value\r\n" );
document.write( "since there is no place to substitute the y\r\n" );
document.write( "value.  That is prefectly OK:\r\n" );
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document.write( "        x > 1\r\n" );
document.write( "       -3 > 1\r\n" );
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document.write( "This is false, so its solutions lie on the\r\n" );
document.write( "OPPOSITE side of the boundary line that the test \r\n" );
document.write( "poin was on.  Therefore we make some marks\r\n" );
document.write( "along the RIGHT side of the green line:\r\n" );
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document.write( " \r\n" );
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document.write( "Now we're ready to draw the next boundary\r\n" );
document.write( "line, x = 1. This is a vertical line that\r\n" );
document.write( "intersects the x axis at 1. Again it\r\n" );
document.write( "should be a dotted line, and you should\r\n" );
document.write( "draw it dotted on your paper:\r\n" );
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document.write( "Now we're ready to draw the last boundary\r\n" );
document.write( "line, y = 3. This is a horizontal line that\r\n" );
document.write( "intersects the y axis at 3. Again it\r\n" );
document.write( "should be a dotted line, and you should\r\n" );
document.write( "draw it dotted on your paper:\r\n" );
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document.write( "We can use the same test point we used for the\r\n" );
document.write( "other two lines since it is not on this line\r\n" );
document.write( "either. So we substitute (-3,5) into the inequality.\r\n" );
document.write( "Not that we need only substitute the y-value\r\n" );
document.write( "since there is no place to substitute the y\r\n" );
document.write( "value.  That is prefectly OK:\r\n" );
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document.write( "        y < 1\r\n" );
document.write( "        5 < 1\r\n" );
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document.write( "This is false, so its solutions lie on the\r\n" );
document.write( "OPPOSITE side of the boundary line that the test \r\n" );
document.write( "poin was on.  Therefore we make some marks\r\n" );
document.write( "along the LOWER side of the blue line:\r\n" );
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document.write( "Now we see that the solution set is the region\r\n" );
document.write( "which is on the same sides of all three lines\r\n" );
document.write( "which the X's are on.  So the triangle region\r\n" );
document.write( "is the solution.  You should shade that solution.\r\n" );
document.write( "I can't shade the triangle on here but you can on\r\n" );
document.write( "your paper.  Also don't forget to draw the lines \r\n" );
document.write( "dotted and not solid.  You should draw them\r\n" );
document.write( "solid only when the inequality is either <\r\n" );
document.write( "or >, but dotted for > or <.\r\n" );
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document.write( "Another comment:  The easiest point to choose\r\n" );
document.write( "as a test point is NOT the point I chose,\r\n" );
document.write( "(-3,5)!!  A much easier test point to choose is\r\n" );
document.write( "the origin (0,0).  The origin may be chosen\r\n" );
document.write( "as a test point any time the boundary line does \r\n" );
document.write( "not pass through the origin.  The reason I did \r\n" );
document.write( "not choose the origin here as a test point was\r\n" );
document.write( "to show you that the choice of test points is \r\n" );
document.write( "really arbitrarily. If I had chosen the origin \r\n" );
document.write( "each time, you might have thought you always \r\n" );
document.write( "had to choose the origin, and would not know\r\n" );
document.write( "what to do if you ran into a problem where a\r\n" );
document.write( "boundary line passed through the origin. \r\n" );
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document.write( "Edwin
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