document.write( "Question 1120991: Solve the below rational inequalities:\r
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Algebra.Com's Answer #736766 by ikleyn(52788) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( " First step of the solution is to transform the inequality to the STANDARD FORM having the rational function in the left side\r
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document.write( " -  <= 0\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( " <= 0\r\n" );
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document.write( " <= 0  \r\n" );
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document.write( " >= 0      (1)\r\n" );
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document.write( "Notice that in the last line I replaced (2-x) in the denominator with (x-2)  and changed the inequality sigh to the opposite one.\r\n" );
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document.write( "Now all the terms in the rational function are of the form (x-c), where \"c\" is the constant, so they are easy to analyse.\r\n" );
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\n" ); document.write( "\n" ); document.write( " First step is done. \r
\n" ); document.write( "\n" ); document.write( " From this point, the standard analysis begins, and it completes the solution.\r
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document.write( "There are 3 critical points,  x= -4,  x= -2/5  and  x= 2.  The domain is the number line except  x= -4  and  x= 2.\r\n" );
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document.write( "The critical points divide the number line in 4 intervals:\r\n" );
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document.write( "    1)  (,)   2)  (,]   3)  [,)   and  4)  (,).\r\n" );
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document.write( "a)  in the first interval, all three binomials of (1) are negative; hence, the rational function (1) is negative.\r\n" );
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document.write( "    Thus this interval  (,)  is NOT the solution.\r\n" );
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document.write( "b)  in the second interval, one of the three binomials of (1) is positive, while the other two are negative.\r\n" );
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document.write( "    Hence, the rational function (1) is positive.\r\n" );
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document.write( "    Thus this interval  (,]  IS the solution.\r\n" );
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document.write( "c)  in the third interval, two of the three binomials of (1) are positive, while the single one is negative.\r\n" );
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document.write( "    Hence, the rational function (1) is negative.\r\n" );
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document.write( "    Thus this interval  [,)  is NOT the solution.\r\n" );
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document.write( "d)  Finally, in the fourth interval, all three binomials of (1) are positive.\r\n" );
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document.write( "    Hence, the rational function (1) is positive.\r\n" );
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document.write( "    Thus this interval  (,)  IS the solution.\r\n" );
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document.write( "Answer.  The solution is the set  (,]  U  (,).\r\n" );
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\n" ); document.write( "\n" ); document.write( "To see many other similar solved problems, look into the lesson\r
\n" ); document.write( "\n" ); document.write( " - Solving inequalities for rational functions with numerator and denominator factored into a product of linear binomials \r
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