document.write( "Question 62776: help please
\n" ); document.write( "solve the rational inequality (x-2)/(x+3) is less than or equal to 0\r
\n" ); document.write( "\n" ); document.write( "is it: (-3,2] (-oo,-3) (-oo,-3)U{2,oo) or [2,oo)
\n" ); document.write( "i am very confused \n" ); document.write( "

Algebra.Com's Answer #854041 by ikleyn(53712) $\"\"$ $\"About$

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\n" ); document.write( "help please
\n" ); document.write( "solve the rational inequality (x-2)/(x+3) is less than or equal to 0
\n" ); document.write( "is it: (-3,2] (-oo,-3) (-oo,-3)U{2,oo) or [2,oo)
\n" ); document.write( "i am very confused
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\n" ); document.write( "\n" ); document.write( "        The solution in the post by @jai_kos in incorrect.\r
\n" ); document.write( "\n" ); document.write( "        It is incorrect methodologically and gives incorrect answer.\r
\n" ); document.write( "\n" ); document.write( "        See my correct solution below.\r
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document.write( "They want you solve this inequality\r\n" );
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document.write( "     <= 0.    (1)\r\n" );
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document.write( "The left side rational function can be non-positive if and only if\r\n" );
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document.write( "    EITHER the numerator is non-positive and denominator is positive\r\n" );
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document.write( "        x - 2 <= 0  and  x + 3 > 0    (2)\r\n" );
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document.write( "    OR     the numerator is non-negative and denominator is negative\r\n" );
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document.write( "        x - 2 >= 0  and  x + 3 < 0.   (3)\r\n" );
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document.write( "In case (2),  x <= 2  and  x > -3  simultaneously, or  \r\n" );
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document.write( "              -3 < x <= 2.\r\n" );
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document.write( "In case (3),  x >= 2  and  x < -3 simultaneously,  which has no solutions.  \r\n" );
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document.write( "Thus the final solution to the given inequality is this set of real numbers  -3 < x <= 2,\r\n" );
document.write( "or, in the interval notation, the set (-3,2].\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "The error made by @jai_kos is that when he multiplies both sides of the original inequality by (x+3),
\n" ); document.write( "he misses the case when (x+3) is negative, which requires different treatment.\r
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\n" ); document.write( "\n" ); document.write( "This error, which jai_kos makes solving the problem, is a typical error, which beginners make
\n" ); document.write( "when trying to solve such inequalities,
\n" ); document.write( "until the more experienced teachers/tutors will explain their error and will show a right way solving.\r
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