document.write( "Question 905763: solve the following inequality: x^2|x-5|>6x. (X squared times the absolute value of x-5 is greater than 6x). Show your work. \n" ); document.write( "
Algebra.Com's Answer #549388 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"x%5E2abs%28x-5%29%3E6x\"\r\n" );
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document.write( "We get 0 on the right\r\n" );
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document.write( "\"x%5E2abs%28x-5%29-6x%3E0\"\r\n" );
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document.write( "Let's find all the critical numbers, which are the\r\n" );
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document.write( "solutions to the equality\r\n" );
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document.write( "\"x%5E2abs%28x-5%29-6x=0\"\r\n" );
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document.write( "That breaks into two cases\r\n" );
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document.write( "(1)  \"x%5E2%28x-5%29-6x=0\"   \r\n" );
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document.write( "and     \r\n" );
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document.write( "(2)  \"x%5E2%28-%28x-5%29%5E%22%22%29-6x=0\"\r\n" );
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document.write( "-----------------\r\n" );
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document.write( "For case (1)\r\n" );
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document.write( "(1)  \"x%5E2%28x-5%29-6x=0\"\r\n" );
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document.write( "     \"x%5E3-5x%5E2-6x=0\"\r\n" );
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document.write( "     \"x%28x%5E2-5x-6%29=0\"\r\n" );
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document.write( "     \"x%28x-6%29%28x%2B1%29=0\"     \r\n" );
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document.write( "    x=0; x-6=0;  x+1=0\r\n" );
document.write( "           x=6;    x=-1  \r\n" );
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document.write( "That gives critical numbers 0, 6, -1\r\n" );
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document.write( "----\r\n" );
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document.write( "For case (2)\r\n" );
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document.write( "(2)  \"x%5E2%28-%28x-5%29%5E%22%22%29-6x=0\"\r\n" );
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document.write( "(2)  \"x%5E2%28-x%2B5%29-6x=0\"\r\n" );
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document.write( "     \"-x%5E3%2B5x%5E2-6x=0\"\r\n" );
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document.write( "     \"x%5E3-5x%5E2%2B6x=0\"\r\n" );
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document.write( "     \"x%28x%5E2-5x%2B6%29=0\"\r\n" );
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document.write( "     \"x%28x-3%29%28x-2%29=0\"     \r\n" );
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document.write( "    x=0; x-3=0;  x-2=0\r\n" );
document.write( "           x=3;    x=2  \r\n" );
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document.write( "That gives critical numbers 0, 3, 2\r\n" );
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document.write( "So from both cases we have critical numbers\r\n" );
document.write( "in order from smallest to largest\r\n" );
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document.write( "-1, 0 , 2, 3, 6\r\n" );
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document.write( "We put those all on a number line\r\n" );
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document.write( "----------o--o-----o--o--------o---------\r\n" );
document.write( "-4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9\r\n" );
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document.write( "We test the original inequality in all intervals\r\n" );
document.write( "between or beyond all those critical numbers.\r\n" );
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document.write( "Left of -1, we choose test value -2, and substitute it in\r\n" );
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document.write( "\"x%5E2abs%28x-5%29%3E6x\"\r\n" );
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document.write( "\"%28-2%29%5E2abs%28-2-5%29%3E6%28-2%29\"\r\n" );
document.write( "\"4abs%28-7%29%3E-12\"\r\n" );
document.write( "\"4%287%29%3E-12\"\r\n" );
document.write( "\"28%3E-12\"\r\n" );
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document.write( "That's true so we shade the part of the number line \r\n" );
document.write( "left of -1\r\n" );
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document.write( "<=========o--o-----o--o--------o---------\r\n" );
document.write( "-4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9\r\n" );
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document.write( "---\r\n" );
document.write( "Between -1 and 0, we choose test value -0.5, and substitute it in\r\n" );
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document.write( "\"x%5E2abs%28x-5%29%3E6x\"\r\n" );
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document.write( "\"%28-0.5%29%5E2abs%28-0.5-5%29%3E6%28-0.5%29\"\r\n" );
document.write( "\"0.25abs%28-5.5%29%3E-3\"\r\n" );
document.write( "\"0.25%285.5%29%3E-3\"\r\n" );
document.write( "\"1.375%3E-12\"\r\n" );
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document.write( "That's also true so we shade the part of the number line \r\n" );
document.write( "between -1 and 0\r\n" );
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document.write( "<=========o==o-----o--o--------o---------\r\n" );
document.write( "-4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9\r\n" );
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document.write( "---\r\n" );
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document.write( "Between 0 and 2, we choose test value 1, and substitute it in\r\n" );
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document.write( "\"x%5E2abs%28x-5%29%3E6x\"\r\n" );
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document.write( "\"%281%29%5E2abs%281-5%29%3E6%281%29\"\r\n" );
document.write( "\"1abs%28-4%29%3E6\"\r\n" );
document.write( "\"4%3E6\"\r\n" );
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document.write( "That's FALSE so we do not shade the part of the number line \r\n" );
document.write( "between 0 and 2.  So we still have:\r\n" );
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document.write( "<=========o==o-----o--o--------o---------\r\n" );
document.write( "-4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9\r\n" );
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document.write( "---\r\n" );
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document.write( "Between 2 and 3, we choose test value 2.5, and substitute it in\r\n" );
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document.write( "\"x%5E2abs%28x-5%29%3E6x\"\r\n" );
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document.write( "\"%282.5%29%5E2abs%282.5-5%29%3E6%282.5%29\"\r\n" );
document.write( "\"6.25abs%28-2.5%29%3E15\"\r\n" );
document.write( "\"6.25%282.5%29%3E15\"\r\n" );
document.write( "\"15.625%3E15\"\r\n" );
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document.write( "That's TRUE so we shade the part of the number line \r\n" );
document.write( "between 2 and 3\r\n" );
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document.write( "<=========o==o-----o==o--------o---------\r\n" );
document.write( "-4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9\r\n" );
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document.write( "---\r\n" );
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document.write( "Between 3 and 4, we choose test value 4, and substitute it in\r\n" );
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document.write( "\"x%5E2abs%28x-5%29%3E6x\"\r\n" );
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document.write( "\"%284%29%5E2abs%284-5%29%3E6%284%29\"\r\n" );
document.write( "\"4abs%28-1%29%3E24\"\r\n" );
document.write( "\"4%281%29%3E6\"\r\n" );
document.write( "\"-4%3E6\"\r\n" );
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document.write( "That's FALSE so we do not shade the part of the number line \r\n" );
document.write( "between 3 and 6.  So we still have:\r\n" );
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document.write( "<=========o==o-----o==o--------o---------\r\n" );
document.write( "-4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9\r\n" );
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document.write( "---\r\n" );
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document.write( "Right of 6, we choose test value 7, and substitute it in\r\n" );
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document.write( "\"x%5E2abs%28x-5%29%3E6x\"\r\n" );
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document.write( "\"%287%29%5E2abs%287-5%29%3E6%287%29\"\r\n" );
document.write( "\"49abs%282%29%3E42\"\r\n" );
document.write( "\"49%282%29%3E42\"\r\n" );
document.write( "\"98%3E42\"\r\n" );
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document.write( "That's true so we shade the part of the number line \r\n" );
document.write( "right of 6\r\n" );
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document.write( "<=========o==o-----o==o--------o========>\r\n" );
document.write( "-4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9\r\n" );
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document.write( "Finally we test the critical numbers:\r\n" );
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document.write( "We test -1\r\n" );
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document.write( "\"%28-1%29%5E2abs%28-1-5%29%3E6%28-1%29\"\r\n" );
document.write( "\"1abs%28-6%29%3E-6\"\r\n" );
document.write( "\"1%286%29%3E-6\"\r\n" );
document.write( "\"6%3E-6\"\r\n" );
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document.write( "So the critical value -1 is a solution so we shade -1\r\n" );
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document.write( "<============o-----o==o--------o========>\r\n" );
document.write( "-4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9\r\n" );
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document.write( "I'll leave it up to you to test the others critical\r\n" );
document.write( "numbers.  You'll find that none of the other critical\r\n" );
document.write( "numbers are solutions by substituting them into the\r\n" );
document.write( "original inequality.  But you should test them.\r\n" );
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document.write( "In set builder notation the solution is:\r\n" );
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document.write( "{x|x<0,26}\r\n" );
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document.write( "In interval notation, the solution is:\r\n" );
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document.write( "(\"-infinity\",0)U(2,3)U(6,\"infinity\") \r\n" );
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document.write( "Edwin

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