Question 1016567
.
How would I solve this rational inequality:
(x/x-6)<2
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<pre>
{{{x/(x-6)}}} < 2.   (1)


1. Assume that x - 6 > 0, i.e. x > 6.
   
   Multiply both sides of (1) by (x-6), which is positive in this case. You will get an inequality

   x < 2*(x-6)  --->  x < 2x - 12  --->  12 > x.

   Thus the solution in this case is the set of real {x | 6 < x < 12}, i.e the interval (6,12).


2. Assume that x - 6 < 0, i.e. x < 6.
   
   Multiply both sides of (1) by (x-6), which is negative in this case. You will get an inequality

   x > 2*(x-6)  --->  x > 2x - 12  --->  12 > x.   <----  Notice that I changed the inequality sign when multiplied by negative number!

   Thus the solution in this case is the set of real {x | x < 6}, i.e the semi-infinite interval ({{{-infinity}}},6).

<U>Answer</U>. The solution is the union of two intervals: ({{{-infinity}}},6) U {6,12}.
</pre>

The plot of the function f(x) = {{{x/(x-6)}}} is shown below.

<TABLE> 
  <TR>
  <TD> 

{{{graph( 330, 330, -6.5, 16.5, -10.5, 10.5,
          x/(x-6)
)}}}


<B>Figure</B>. Plot y = {{{x/(x-6)}}}

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