document.write( "Question 1209289: Solve the inequality (x - 2)(x + 6) \le -x^2 + 5x - 15. Write your answer in interval notation. \n" ); document.write( "

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\n" ); document.write( "Solve the inequality (x - 2)(x + 6) \le -x^2 + 5x - 15. Write your answer in interval notation.
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document.write( "Starting inequality is \r\n" );
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document.write( "    (x - 2)(x + 6) <= -x^2 + 5x - 15.\r\n" );
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document.write( "Simplify and reduce to the standard form quadratic equation\r\n" );
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document.write( "    x^2 + 4x - 12 <= -x^2 + 5x - 15,\r\n" );
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document.write( "    2x^2 - x + 3 <= 0.\r\n" );
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document.write( "Calculate the discriminant\r\n" );
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document.write( "    d = b^2 - 4ac = (-1)^2 - 4*2*3 = 1 - 24 = -23.\r\n" );
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document.write( "The discriminant is negative.\r\n" );
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document.write( "It means that the quadratic function  y = 2x^2 - x + 3  has no zeroes and is positive \r\n" );
document.write( "everywhere on the number line for all real numbers.\r\n" );
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document.write( "Therefore, the set of solutions is EMPTY.  \r\n" );
document.write( "It can not be presented in the interval notation.\r\n" );
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