document.write( "Question 1052325: Find the range of values of 'c' such that the two lines
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Algebra.Com's Answer #667702 by Edwin McCravy(20060)\"\" \"About 
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Find the range of values of 'c' such that the two lines
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All points in the first quadrant have both\r\n" );
document.write( "their x and y coordinates positive.\r\n" );
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document.write( "   x - y = 2\r\n" );
document.write( "  cx + y = 3\r\n" );
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document.write( "Adding the two equations term by term:\r\n" );
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document.write( "  x + cx = 5\r\n" );
document.write( "  x(1+c) = 5\r\n" );
document.write( "       x = 5/(1+c)\r\n" );
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document.write( "This must be positive, so 1+c > 0 or c > -1\r\n" );
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document.write( "Substitute in the first:\r\n" );
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document.write( "   x - y = 2\r\n" );
document.write( "      -y = 2 - x\r\n" );
document.write( "       y = -2 + x\r\n" );
document.write( "       y = -2 + 5/(1+c)\r\n" );
document.write( "       y = -2(1+c)/(1+c) + 5/(1+c)\r\n" );
document.write( "       y = (-2-2c)/(1+c) + 5/(1+c)\r\n" );
document.write( "       y = (-2-2c+5)/(1+c)\r\n" );
document.write( "       y = (3-2c)/(1+c)\r\n" );
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document.write( "This must be positive\r\n" );
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document.write( "\"%283-2c%29%2F%281%2Bc%29%3E0\"\r\n" );
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document.write( "This has critical numbers which are the\r\n" );
document.write( "zeros of numerator and denominator.\r\n" );
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document.write( "Find the zero of the numerator:\r\n" );
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document.write( "3-2c=0 \r\n" );
document.write( " -2c=-3\r\n" );
document.write( "   c=3/2 = 1.5\r\n" );
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document.write( "Find the zero of the denominator:\r\n" );
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document.write( "1+c=0\r\n" );
document.write( "  c=-1\r\n" );
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document.write( "We place the critical numbers on a number line:\r\n" );
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document.write( "---------o---------o----------\r\n" );
document.write( "-3  -2  -1   0   1   2   3   4   \r\n" );
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document.write( "Choose a test point -2 in the leftmost interval\r\n" );
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document.write( "\"%283-2c%29%2F%281%2Bc%29%3E0\"\r\n" );
document.write( "\"%283-2%28-2%29%29%2F%281%2B%28-2%29%29%3E0\"\r\n" );
document.write( "\"%283%2B4%29%2F%281-2%29%3E0\"\r\n" );
document.write( "\"7%2F%28-1%29%3E0\"\r\n" );
document.write( "\"-7%3E0\"\r\n" );
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document.write( "This is false so the interval \"%28matrix%281%2C3%2C-infinity%2C%22%2C%22%2C-1%29%29\"\r\n" );
document.write( "is not included in the range of values for c.\r\n" );
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document.write( "---------o---------o----------\r\n" );
document.write( "-3  -2  -1   0   1   2   3   4   \r\n" );
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document.write( "Choose a test point 0 in the middle interval\r\n" );
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document.write( "\"%283-2c%29%2F%281%2Bc%29%3E0\"\r\n" );
document.write( "\"%283-2%280%29%29%2F%281%2B%280%29%29%3E0\"\r\n" );
document.write( "\"%283%29%2F%281%29%3E0\"\r\n" );
document.write( "\"3%3E0\"\r\n" );
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document.write( "This is true so the interval \"%28matrix%281%2C3%2C-1%2C%22%2C%22%2C3%2F2%29%29\"\r\n" );
document.write( "is included in the range of values for c.\r\n" );
document.write( "\r\n" );
document.write( "---------o---------o----------\r\n" );
document.write( "-3  -2  -1   0   1   2   3   4   \r\n" );
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document.write( "Choose a test point 2 in the rightmost interval\r\n" );
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document.write( "\"%283-2c%29%2F%281%2Bc%29%3E0\"\r\n" );
document.write( "\"%283-2%282%29%29%2F%281%2B%282%29%29%3E0\"\r\n" );
document.write( "\"%283-4%29%2F%281%2B2%29%3E0\"\r\n" );
document.write( "\"%28-1%29%2F3%3E0\"\r\n" );
document.write( "\"-1%2F3%3E0\"\r\n" );
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document.write( "This is false so the interval \"%28matrix%281%2C3%2C3%2F2%2C%22%2C%22%2Cinfinity%29%29\"\r\n" );
document.write( "is not included in the range of values for c.\r\n" );
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document.write( "Answer: \"%28matrix%281%2C3%2C-1%2C%22%2C%22%2C3%2F2%29%29\"\r\n" );
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document.write( "Edwin
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