document.write( "Question 878371: The solution of the inequality sqrt( 5-x ) < x+1 is (a) < x ≤ (B) \r
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Algebra.Com's Answer #529916 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"sqrt%285-x%29\"\"%22%22%3C%22%22\"\"x%2B1\"\r\n" );
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document.write( "Get 0 on the right.\r\n" );
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document.write( "\"sqrt%285-x%29-x-1\"\"%22%22%3C%22%22\"\"%220%22\"\r\n" );
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document.write( "In order for \"sqrt%285-x%29\" to be a real number we require\r\n" );
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document.write( "\"5-x\"\"%22%22%3E=%22%22\"\"%220%22\"\r\n" );
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document.write( "\"5\"\"%22%22%3E=%22%22\"\"x\" which is the same as   \r\n" );
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document.write( "\"x\"\"%22%22%3C=%22%22\"\"5\"\r\n" );
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document.write( "So 5 is a critical number, since no value of x\r\n" );
document.write( "can be greater than 5\r\n" );
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document.write( "Next we look for critical numbers of\r\n" );
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document.write( "\"sqrt%285-x%29-x-1\"\"%22%22%3C%22%22\"\"%220%22\"\r\n" );
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document.write( "By solving \"sqrt%285-x%29-x-1\"\"%22%22=%22%22\"\"%220%22\"\r\n" );
document.write( "           \r\n" );
document.write( "           \"sqrt%285-x%29\"\"%22%22=%22%22\"\"x%2B1\"\r\n" );
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document.write( "           \"%28sqrt%285-x%29%29%5E2\"\"%22%22=%22%22\"\"%28x%2B1%29%5E2\"\r\n" );
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document.write( "           \"5-x\"\"%22%22=%22%22\"\"x%5E2%2B2x%2B1\"\r\n" );
document.write( "  \r\n" );
document.write( "           \"%220%22\"\"%22%22=%22%22\"\"x%5E2%2B3x-4\"\r\n" );
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document.write( "           \"%220%22\"\"%22%22=%22%22\"\"%28x%2B4%29%28x-1%29\"\r\n" );
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document.write( "            x+4=0;  x-1=0\r\n" );
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document.write( "              x=-4;   x=1\r\n" );
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document.write( "So there are two more critical numbers\r\n" );
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document.write( "We plot the three critical numbers on a number line:\r\n" );
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document.write( "----------o--------------o-----------o\r\n" );
document.write( "-7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5\r\n" );
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document.write( "We test a value in each of the three regions\r\n" );
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document.write( "For the leftmost region, we test x=-5 in \r\n" );
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document.write( "\"sqrt%285-x%29-x-1\"\"%22%22%3C%22%22\"\"%220%22\"\r\n" );
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document.write( "\"sqrt%285-%28-5%29%29-%28-5%29-1\"\"%22%22%3C%22%22\"\"%220%22\"\r\n" );
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document.write( "\"sqrt%2810%29%2B5-1\"\"%22%22%3C%22%22\"\"%220%22\"\r\n" );
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document.write( "\"sqrt%2810%29%2B4\"\"%22%22%3C%22%22\"\"%220%22\"\r\n" );
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document.write( "The left side is a positive number, so that is false.\r\n" );
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document.write( "So the left region is not part of the solution:\r\n" );
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document.write( "----------o--------------o-----------o\r\n" );
document.write( "-7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5\r\n" );
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document.write( "For the middle region, we test x=0 in \r\n" );
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document.write( "\"sqrt%285-x%29-x-1\"\"%22%22%3C%22%22\"\"%220%22\"\r\n" );
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document.write( "\"sqrt%285-%280%29%29-%280%29-1\"\"%22%22%3C%22%22\"\"%220%22\"\r\n" );
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document.write( "\"sqrt%285%29-1\"\"%22%22%3C%22%22\"\"%220%22\"\r\n" );
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document.write( "The left side is a positive number, so that is false.\r\n" );
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document.write( "So the middle region is also not part of the solution:\r\n" );
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document.write( "----------o--------------o-----------o\r\n" );
document.write( "-7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5\r\n" );
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document.write( "For the righmost region, we test x=2 in \r\n" );
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document.write( "\"sqrt%285-x%29-x-1\"\"%22%22%3C%22%22\"\"%220%22\"\r\n" );
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document.write( "\"sqrt%285-%282%29%29-%282%29-1\"\"%22%22%3C%22%22\"\"%220%22\"\r\n" );
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document.write( "\"sqrt%283%29-3\"\"%22%22%3C%22%22\"\"%220%22\"\r\n" );
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document.write( "The left side is a negative number, so that is true.\r\n" );
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document.write( "So the rightmost region is part of the solution,\r\n" );
document.write( "so we shade that interval:\r\n" );
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document.write( "----------o--------------o===========o\r\n" );
document.write( "-7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5\r\n" );
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document.write( "Finally we test the critical numbers themselves to\r\n" );
document.write( "see if they are solutions;\r\n" );
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document.write( "We test critical number -4 in \r\n" );
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document.write( "\"sqrt%285-x%29-x-1\"\"%22%22%3C%22%22\"\"%220%22\"\r\n" );
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document.write( "\"sqrt%285-%28-4%29%29-%28-4%29-1\"\"%22%22%3C%22%22\"\"%220%22\"\r\n" );
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document.write( "\"sqrt%285%2B4%29%2B4-1\"\"%22%22%3C%22%22\"\"%220%22\"\r\n" );
document.write( "    \r\n" );
document.write( "\"sqrt%289%29%2B3\"\"%22%22%3C%22%22\"\"%220%22\"\r\n" );
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document.write( "\"3%2B3\"\"%22%22%3C%22%22\"\"%220%22\"\r\n" );
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document.write( "\"6\"\"%22%22%3C%22%22\"\"%220%22\"\r\n" );
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document.write( "That's false so critical number -4 is not a solution.\r\n" );
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document.write( "So we erase the open circle at -4:\r\n" );
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document.write( "-------------------------o===========o\r\n" );
document.write( "-7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5\r\n" );
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document.write( "We test critical number 1 in \r\n" );
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document.write( "\"sqrt%285-x%29-x-1\"\"%22%22%3C%22%22\"\"%220%22\"\r\n" );
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document.write( "\"sqrt%285-%281%29%29-%281%29-1\"\"%22%22%3C%22%22\"\"%220%22\"\r\n" );
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document.write( "\"sqrt%284%29-2\"\"%22%22%3C%22%22\"\"%220%22\"\r\n" );
document.write( "    \r\n" );
document.write( "\"2-2\"\"%22%22%3C%22%22\"\"%220%22\"\r\n" );
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document.write( "\"%220%22\"\"%22%22%3C%22%22\"\"%220%22\"\r\n" );
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document.write( "That's false so critical number 1 is not a solution.\r\n" );
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document.write( "-------------------------o===========o\r\n" );
document.write( "-7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5\r\n" );
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document.write( "We test critical number 5 in \r\n" );
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document.write( "\"sqrt%285-x%29-x-1\"\"%22%22%3C%22%22\"\"%220%22\"\r\n" );
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document.write( "\"sqrt%285-%285%29%29-%285%29-1\"\"%22%22%3C%22%22\"\"%220%22\"\r\n" );
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document.write( "\"sqrt%280%29-6\"\"%22%22%3C%22%22\"\"%220%22\"\r\n" );
document.write( "    \r\n" );
document.write( "\"-6\"\"%22%22%3C%22%22\"\"%220%22\"\r\n" );
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document.write( "That's true so 5 is a solution.\r\n" );
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document.write( "So we darken it:\r\n" );
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document.write( "-------------------------o===========●\r\n" );
document.write( "-7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5\r\n" );
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document.write( "So the solution is \r\n" );
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document.write( "1 < x ≤ 5\r\n" );
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document.write( "That can be written in interval notation as (1,5]\r\n" );
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document.write( "Edwin
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