Question 1018478
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How would I solve (3x+2/x+1) > 4 ? Thank you for your help
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{{{(3x+2)/(x+1)}}} > {{{4}}}.       (1)


Now see how it <U>SHOULD</U> be done.


<pre>
1. First, let us assume that x+1 > 0.
   In other words, we will consider now real numbers { x | x > -1 }. 

   Multiply both side of (1) by (x+1), which is positive in this case. Then you will get

   3x+2 > 4*(x+1)  --->  3x+2 > 4x+4  --->  2-4 > 4x-3x  --->  -2 > x.

   Thus we obtain this: if x > -1, then x < -2. 
   
   It is, surely, absurd. 
   So, in the domain x > -1 there is no solution to (1).


2. Next, let us consider the interval x < -1. In this interval, the denominator (x+1) is negative.

   Multiply both side of (1) by (x+1), which is negative now. Then you will get

   3x+2 < 4*(x+1).     (2) 

   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).

   Further, (2) implies  3x+2 < 4x+4  --->  2-4 < 4x-3x  --->  -2 < x,   or   x > -2.
   
   Thus we obtain this: if x < -1, then x > -2.

   It means that the set of real numbers -2 < x < -1 satisfies the inequality (1).

   It is the solution of the inequality (1).

<U>Answer</U>. The solution to (1) is the interval (-2,-1).
</pre>

Below is the plot, for illustration.

<TABLE> 
  <TR>
  <TD>

{{{graph( 330, 330, -5.5, 5.5, -5.5, 10.5,
          (3x+2)/(x+1), 4
)}}}


&nbsp;&nbsp;&nbsp;&nbsp;<B>Figure 1</B>. Plot y = {{{(3x+2)/(x+1)}}}

  </TD>
  </TR>
</TABLE>


For similar problems, see the lesson <A HREF=http://www.algebra.com/algebra/homework/Inequalities/Solving-inequalities-for-rational-functions-with-non-zero-right-side.lesson>Solving inequalities for rational functions with non-zero right side</A> in this site.