Question 1019514
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How would this be solved: {{{(3x+1)/(x-2) > 2}}}
Thank you
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{{{(3x+1)/(x-2))}}} > 2.      (1)


1. First, let us assume that x-2 > 0.
   In other words, we will look now for solutions in the domain  { x | x > 2 }. 

   Multiply both side of (1) by (x-2), which is positive in this case. Then you will get inequality&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

   3x+1 > 2*(x-2)  --->  3x+1 > 2x-4  --->  3x - 2x > -4 -1  --->  x > -5.

   Thus we obtain this: the solution is the intersection of two sets: {x| x > 2} and {x| x > -5}. 
   
   This intersection is the set {x| x > 2}.
   
   So, the interval ({{{2}}},{{{infinity}}}) is the solution to (1), under the assumption that x > 2.


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

   Multiply both side of (1) by (x-2), which is negative now. Then you will get

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

   Notice, that I changed the sign ">" of the original inequality to the opposite sign "<", when 
   multiplied both sides of (1) by negative number (x-2).

   Now, (2) implies  3x+1 < 2x-4  --->  3x - 2x < -4 -1  --->  x < -5.
   
   Thus by analyzing the domain x < 2 we obtain the solution  x < -5.

   By collecting the results of n.1 and n.2 you get the full solution set.
   It is the union ({{{-infinity}}},{{{-5}}}) U ({{{2}}},{{{infinity}}}).

   The problem is solved.

<U>Answer</U>. The solution to  (1)  is the union  ({{{-infinity}}},{{{-5}}}) U ({{{2}}},{{{infinity}}}).


The plot of the function  {{{(3x+1)/(x-2)}}}  is shown in <B>Figure</B>.

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


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

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