document.write( "Question 771814: |X-1|+|X-2|+|X-4|>=6 \n" ); document.write( "
Algebra.Com's Answer #470497 by Edwin McCravy(20056)\"\" \"About 
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document.write( "|x-1| + |x-2| + |x-4| ≧ 6\r\n" );
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document.write( "There are 8 cases to consider\r\n" );
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document.write( "1. x-1 ≧ 0, x-2 ≧ 0, and x-4 ≧ 0\r\n" );
document.write( "2. x-1 ≧ 0, x-2 ≧ 0, and x-4 ≦ 0\r\n" );
document.write( "3. x-1 ≧ 0, x-2 ≦ 0, and x-4 ≧ 0\r\n" );
document.write( "4. x-1 ≧ 0, x-2 ≦ 0, and x-4 ≦ 0\r\n" );
document.write( "5. x-1 ≦ 0, x-2 ≧ 0, and x-4 ≦ 0\r\n" );
document.write( "6. x-1 ≦ 0, x-2 ≧ 0, and x-4 ≦ 0\r\n" );
document.write( "7. x-1 ≦ 0, x-2 ≦ 0, and x-4 ≧ 0\r\n" );
document.write( "8. x-1 ≦ 0, x-2 ≦ 0, and x-4 ≦ 0\r\n" );
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document.write( "They simplify to\r\n" );
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document.write( "1. x ≧ 1, x ≧ 2, and x ≧ 4 ↠ x ≧ 4\r\n" );
document.write( "2. x ≧ 1, x ≧ 2, and x ≦ 4 ↠ 2 ≦ x ≦ 4\r\n" );
document.write( "3. x ≧ 1, x ≦ 2, and x ≧ 4 ↠ contradiction\r\n" );
document.write( "4. x ≧ 1, x ≦ 2, and x ≦ 4 ↠ 1 ≦ x ≦ 2\r\n" );
document.write( "5. x ≦ 1, x ≧ 2, and x ≦ 4 ↠ contradiction\r\n" );
document.write( "6. x ≦ 1, x ≧ 2, and x ≦ 4 ↠ contradiction\r\n" );
document.write( "7. x ≦ 1, x ≦ 2, and x ≧ 4 ↠ contradiction\r\n" );
document.write( "8. x ≦ 3, x ≦ 2, and x ≦ 4 ↠ x ≦ 2\r\n" );
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document.write( "Four of those are obviously contradictions\r\n" );
document.write( "So we have only four cases to consider: \r\n" );
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document.write( "1. x ≧ 1, x ≧ 2, and x ≧ 4 ↠ x ≧ 4\r\n" );
document.write( "2. x ≧ 1, x ≧ 2, and x ≦ 4 ↠ 2 ≦ x ≦ 4\r\n" );
document.write( "4. x ≧ 1, x ≦ 2, and x ≦ 4 ↠ 1 ≦ x ≦ 2\r\n" );
document.write( "8. x ≦ 3, x ≦ 2, and x ≦ 4 ↠ x ≦ 2\r\n" );
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document.write( "Case 1: x ≧ 1, x ≧ 2, and x ≧ 4 ↠ x ≧ 4\r\n" );
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document.write( "1. x-1 ≧ 0, x-2 ≧ 0, and x-4 ≧ 0\r\n" );
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document.write( "  |x-1| + |x-2| + |x-4| ≧ 6 \r\n" );
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document.write( "becomes:\r\n" );
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document.write( "   x-1  +  x-2  +  x-4  ≧ 6\r\n" );
document.write( "                  3x-7  ≧ 6\r\n" );
document.write( "                    3x  ≧ 13\r\n" );
document.write( "                     x  ≧ \"13%2F3\"\r\n" );
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document.write( "Case 2: x ≧ 1, x ≧ 2, and x ≦ 4 ↠ 2 ≦ x ≦ 4\r\n" );
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document.write( "2. x-1 ≧ 0, x-2 ≧ 0, and x-4 ≦ 0\r\n" );
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document.write( "   |x-1| + |x-2| + |x-4| ≧ 6 \r\n" );
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document.write( "becomes:\r\n" );
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document.write( "   x-1  +  x-2  -  x+4  ≧ 6\r\n" );
document.write( "                   x+1  ≧ 6\r\n" );
document.write( "                     x  ≧ 5\r\n" );
document.write( "                       \r\n" );
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document.write( "That contradicts x ≦ 4 so case 2 is eliminated\r\n" );
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document.write( "Case 4: x-1 ≧ 0, x-2 ≦ 0, and x-4 ≦ 0\r\n" );
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document.write( " 4. x ≧ 1, x ≦ 2, and x ≦ 4 ↠ 1 ≦ x ≦ 2\r\n" );
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document.write( "   |x-1| + |x-2| + |x-4| ≧ 6 \r\n" );
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document.write( "beomes:\r\n" );
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document.write( "    x-1  -  x+2  -  x+4  ≧ 6\r\n" );
document.write( "                   -x+6  ≧ 6\r\n" );
document.write( "                     -x  ≧ 0\r\n" );
document.write( "                      x  ≦ 0\r\n" );
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document.write( "That contradicts x ≧ 1 so case 4 is eliminated\r\n" );
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document.write( "Case 8: x-1 ≦ 0, x-2 ≦ 0, and x-4 ≦ 0\r\n" );
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document.write( " 4. x ≦ 1, x ≦ 2, and x ≦ 4 ↠ x ≦ 1\r\n" );
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document.write( "   |x-1| + |x-2| + |x-4| ≧ 6  \r\n" );
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document.write( "becomes:\r\n" );
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document.write( "        -x+1 - x+2 - x+4 ≧ 6\r\n" );
document.write( "                  -3x+7  ≧ 6\r\n" );
document.write( "                    -3x  ≧ -1\r\n" );
document.write( "                      x  ≦ \"1%2F3\"\r\n" );
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document.write( "So the two cases that aren't self- contradictory are\r\n" );
document.write( "cases 1 and 8. So the original inequality is true when\r\n" );
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document.write( "           x  ≦ \"1%2F3\"  or when  x  ≧ \"13%2F3\"\r\n" );
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document.write( "The solution is the union of the solutions for those two cases:\r\n" );
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document.write( "         (-∞,\"1%2F3\") U (\"13%2F3\",∞)\r\n" );
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document.write( "Edwin
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