document.write( "Question 981121: -2 < x^2-4x+1 < 1 \n" ); document.write( "
Algebra.Com's Answer #602173 by Edwin McCravy(20081)\"\" \"About 
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document.write( "-2 < x²-4x+1 < 1\r\n" );
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document.write( "This is the same as\r\n" );
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document.write( "-2 < x²-4x+1    AND   x²-4x+1 < 1 \r\n" );
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document.write( "We find the critical numbers for each\r\n" );
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document.write( "First we find the critical numbers for\r\n" );
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document.write( "-2 < x²-4x+1 which is the same as\r\n" );
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document.write( "x²-4x+1 > -2\r\n" );
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document.write( "x²-4x+3 > 0\r\n" );
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document.write( "(x-1)(x-3) > 0\r\n" );
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document.write( "Critical numbers are 1 and 3\r\n" );
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document.write( "Next we find the critical numbers for\r\n" );
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document.write( "x²-4x+1 < 1\r\n" );
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document.write( "  x²-4x < 0\r\n" );
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document.write( " x(x-4) < 0\r\n" );
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document.write( "Critical numbers are 0 and 4\r\n" );
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document.write( "We place all four critical numbers on a number line:\r\n" );
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document.write( "----------o--o-----o--o---------\r\n" );
document.write( "-3 -2 -1  0  1  2  3  4  5  6  7\r\n" );
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document.write( "We test the interval x < 0 with test value -1\r\n" );
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document.write( "-2 < x²-4x+1 < 1\r\n" );
document.write( "-2 < (-1)²-4(-1)+1 < 1\r\n" );
document.write( "-2 < 1+4+1 < 1\r\n" );
document.write( "-2 < 6 < 1\r\n" );
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document.write( "False so we don't include x < 0\r\n" );
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document.write( "----------o--o-----o--o---------\r\n" );
document.write( "-3 -2 -1  0  1  2  3  4  5  6  7\r\n" );
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document.write( "We test the interval 0 < x < 1 with test value 0.5\r\n" );
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document.write( "-2 < x²-4x+1 < 1\r\n" );
document.write( "-2 < (0.5)²-4(0.5)+1 < 1\r\n" );
document.write( "-2 < 0.25-2+1 < 1\r\n" );
document.write( "-2 < -0.75 < 1\r\n" );
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document.write( "That's true so we include 0 < x < 1\r\n" );
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document.write( "----------o==o-----o--o---------\r\n" );
document.write( "-3 -2 -1  0  1  2  3  4  5  6  7\r\n" );
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document.write( "We test the interval 1 < x < 3 with test value 2\r\n" );
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document.write( "-2 < x²-4x+1 < 1\r\n" );
document.write( "-2 < (2)²-4(2)+1 < 1\r\n" );
document.write( "-2 < 4-8+1 < 1\r\n" );
document.write( "-2 < -3 < 1\r\n" );
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document.write( "That's false so we do not include 1 < x < 3\r\n" );
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document.write( "----------o==o-----o--o---------\r\n" );
document.write( "-3 -2 -1  0  1  2  3  4  5  6  7\r\n" );
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document.write( "We test the interval 3 < x < 4 with test value 3.5\r\n" );
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document.write( "-2 < x²-4x+1 < 1\r\n" );
document.write( "-2 < (3.5)²-4(3.5)+1 < 1\r\n" );
document.write( "-2 < 12.25-14+1 < 1\r\n" );
document.write( "-2 < -0.75 < 1\r\n" );
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document.write( "That's true so we include 3 < x < 4\r\n" );
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document.write( "----------o==o-----o==o---------\r\n" );
document.write( "-3 -2 -1  0  1  2  3  4  5  6  7\r\n" );
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document.write( "We test the interval x > 4 with test value 5\r\n" );
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document.write( "-2 < x²-4x+1 < 1\r\n" );
document.write( "-2 < (5)²-4(5)+1 < 1\r\n" );
document.write( "-2 < 25-20+1 < 1\r\n" );
document.write( "-2 < 6 < 1\r\n" );
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document.write( "That's false so we do not include x > 4\r\n" );
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document.write( "-------------------\r\n" );
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document.write( "Since the inequality is STRICT inequality, the critical numbers\r\n" );
document.write( "cannot be solutions, and the graph of the solution set is\r\n" );
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document.write( "----------o==o-----o==o---------\r\n" );
document.write( "-3 -2 -1  0  1  2  3  4  5  6  7\r\n" );
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document.write( "which in interval notation is:\r\n" );
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document.write( "(0,1) U (3,4)\r\n" );
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document.write( "Edwin
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