document.write( "Question 673800: Solve the following inequality:\r
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Algebra.Com's Answer #418915 by Edwin McCravy(20056) $\"\"$ $\"About$

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$\"+sqrt%28%281%2Fabs%28x%29%29-%281%2Fx%5E2%29%29-%281%2Fx%29\"$ > 0
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document.write( "We have to consider two cases: \r\n" );
document.write( "1. when x > 0 and \r\n" );
document.write( "2. when x < 0 [Note that x cannot equal 0]\r\n" );
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document.write( "Case 1:   x > 0\r\n" );
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document.write( " > 0\r\n" );
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document.write( "We isolate the radical:\r\n" );
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document.write( " > \r\n" );
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document.write( "since x > 0 we may replace |x| by x\r\n" );
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document.write( " >   \r\n" );
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document.write( "The left side is non-negative and the right side is positive.\r\n" );
document.write( "Since the left side is greater than the right side, both sides \r\n" );
document.write( "are positive, and we can square both sides and the inequality \r\n" );
document.write( "will not change:\r\n" );
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document.write( " -  > \r\n" );
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document.write( "Add  to both sides:\r\n" );
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document.write( " > \r\n" );
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document.write( "Multiply through by x² which is positive and \r\n" );
document.write( "so the inequality will not change:\r\n" );
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document.write( "        x > 2\r\n" );
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document.write( "So the only positive values of x for which the original\r\n" );
document.write( "inequality holds is when x is greater than 2.\r\n" );
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document.write( "-------------------------------------\r\n" );
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document.write( "Case 2:   x < 0\r\n" );
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document.write( " > 0\r\n" );
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document.write( "We isolate the radical: \r\n" );
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document.write( " > \r\n" );
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document.write( "Then the right side is negative and the left side\r\n" );
document.write( "is non-negative, so the above inequality will always\r\n" );
document.write( "be true as long as the expression under the radical\r\n" );
document.write( "is not negative.  So we only need require that the\r\n" );
document.write( "expression under the square root radical not be\r\n" );
document.write( "negative.\r\n" );
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document.write( " -  ≥ 0 \r\n" );
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document.write( "In this case |x| = -x and we have\r\n" );
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document.write( " -  ≥ 0\r\n" );
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document.write( " -  ≥ 0\r\n" );
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document.write( "Multiply through by x² which is positive and the\r\n" );
document.write( "inequality will not change:\r\n" );
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document.write( "-x - 1 ≥ 0\r\n" );
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document.write( "Now we'll multiply through by -1 and the inequality\r\n" );
document.write( "WILL change:\r\n" );
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document.write( " x + 1 ≤ 0\r\n" );
document.write( "     x ≤ -1\r\n" );
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document.write( "So the only negative values of x for which the original\r\n" );
document.write( "inequality holds is when x is less than or equal to -1.\r\n" );
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document.write( "-------------------------\r\n" );
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document.write( "Therefore the solution in interval notation is:\r\n" );
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document.write( "(-∞,-1] ⋃ (2,∞) \r\n" );
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document.write( "Edwin

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