Algebra.Com's Answer #731004 by ikleyn(52780)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( " <=  (1)\r\n" );
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document.write( " - <=  \r\n" );
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document.write( " -  <= (written with the common denominator 2x )\r\n" );
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document.write( " <= \r\n" );
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document.write( " <=  (2)\r\n" );
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document.write( "On the left side, we have a rational function of the three factors (2-x), (2+x) and (2x).\r\n" );
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document.write( "There are three critical values, where the factor become equal to zero and change their sign: -2, 0, and 2.\r\n" );
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document.write( "And there are four interval to analyse: a) x < -2; b) -2 <= x < 0; c) 0 < x < 2; and d) x >= 2.\r\n" );
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document.write( "In the interval a) x < - 2 factors (x+2) and (2x) are negative; factor (2-x) is positive. So inequality (2) is FALSE.\r\n" );
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document.write( "In the interval b) -2 <= x < 0 the factor (x+2) is positive, factor (2x) is negative; factor (2-x) is positive. \r\n" );
document.write( "So, the whole function in the left side of (2) is negative, and the inequality (2) IS TRUE.\r\n" );
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document.write( "In the interval c) 0 < x < 2 the factors (x+2) and (2x) are positive; the factor (x-2) is positive; so inequality (2) is FALSE.\r\n" );
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document.write( "In the interval d) x >= 2 factors (x+2) and (2x) are positive; factor (2-x) is negative. So, the whole function in the left side of (2) is negative, and the inequality (2) IS TRUE.\r\n" );
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document.write( "Thus the inequality (2) is TRUE in these two intervals\r\n" );
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document.write( " -2 <= x < 0 and x >= 2.\r\n" );
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document.write( "Since inequalities (1) and (2) are EQUIVALENT (!), your answer is:\r\n" );
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document.write( "Answer. The original inequality is true at -2 <= x < 0 and x >= 2.\r\n" );
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document.write( " The solution is the set [ ,) U [ , ].\r\n" );
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document.write( "Solved.\r
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document.write( "To see many other similar solved problems for inequalities for rational functions, look into the lesson\r
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document.write( " - Solving inequalities for rational functions with numerator and denominator factored into a product of linear binomials \r
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document.write( "in this site.\r
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document.write( "Also, you have this free of charge online textbook in ALGEBRA-I in this site\r
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document.write( " - ALGEBRA-I - YOUR ONLINE TEXTBOOK.\r
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document.write( "The referred lesson is the part of this online textbook under the topic \"Inequalities\".\r
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document.write( "Save the link to this online textbook together with its description\r
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document.write( "Free of charge online textbook in ALGEBRA-I
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document.write( "https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson\r
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document.write( "to your archive and use it when it is needed.\r
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