document.write( "Question 1018478: How would I solve (3x+2/x+1) > 4 ? Thank you for your help \n" ); document.write( "

Algebra.Com's Answer #634584 by ikleyn(52805) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"%283x%2B2%29%2F%28x%2B1%29\"$ > $\"4\"$ . (1)\r
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\n" ); document.write( "\n" ); document.write( "Now see how it SHOULD be done.\r
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document.write( "1. First, let us assume that x+1 > 0.\r\n" );
document.write( "   In other words, we will consider now real numbers { x | x > -1 }. \r\n" );
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document.write( "   Multiply both side of (1) by (x+1), which is positive in this case. Then you will get\r\n" );
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document.write( "   3x+2 > 4*(x+1)  --->  3x+2 > 4x+4  --->  2-4 > 4x-3x  --->  -2 > x.\r\n" );
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document.write( "   Thus we obtain this: if x > -1, then x < -2. \r\n" );
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document.write( "   It is, surely, absurd. \r\n" );
document.write( "   So, in the domain x > -1 there is no solution to (1).\r\n" );
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document.write( "2. Next, let us consider the interval x < -1. In this interval, the denominator (x+1) is negative.\r\n" );
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document.write( "   Multiply both side of (1) by (x+1), which is negative now. Then you will get\r\n" );
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document.write( "   3x+2 < 4*(x+1).     (2) \r\n" );
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document.write( "   Notice, that I changed the sign \">\" of the inequality to the opposite sign \"<\", when I multiplied both sides of (1) by negative number (x+1).\r\n" );
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document.write( "   Further, (2) implies  3x+2 < 4x+4  --->  2-4 < 4x-3x  --->  -2 < x,   or   x > -2.\r\n" );
document.write( "   \r\n" );
document.write( "   Thus we obtain this: if x < -1, then x > -2.\r\n" );
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document.write( "   It means that the set of real numbers -2 < x < -1 satisfies the inequality (1).\r\n" );
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document.write( "   It is the solution of the inequality (1).\r\n" );
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document.write( "Answer. The solution to (1) is the interval (-2,-1).\r\n" );
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\n" ); document.write( "\n" ); document.write( " Figure 1. Plot y = $\"%283x%2B2%29%2F%28x%2B1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "For similar problems, see the lesson Solving inequalities for rational functions with non-zero right side in this site.\r
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