Question 1016566
.
How would I solve the following rational inequality:
(x-5/3x) < 3
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<pre>
{{{(x-5)/(3x)}}} < 3.   (1)


1. Assume that x > 0.
   
   Multiply both sides of (1) by 3x, which is positive in this case. You will get an inequality

  x - 5 < 9x  --->  -5 < 8x  --->  x > {{{-5/8}}}.

  Thus he solution in this case is the set of real {x | x > 0}, i.e the interval ({{{0}}},{{{infinity}}}).


2. Assume that x < 0.
   
   Multiply both sides of (1) by 3x, which is negative in this case. You will get an inequality

  x -5 > 9x  --->  -5 > 8x  --->  x < {{{-5/8}}}.   <----  Notice that I changed the inequality sign when multiplied by negative number!

  Thus he solution in this case is the set of real {x | x < {{{-5/8}}} }, i.e the semi-infinite interval ({{{-infinity}}},{{{-5/8}}}).

<U>Answer</U>. The solution is the union of two intervals: ({{{-infinity}}},{{{-5/8}}}) U ({{{0}}},{{{infinity}}}).
</pre>

The plot of the function f(x) = {{{(x-5)/(3x)}}} is shown below.

<TABLE> 
  <TR>
  <TD> 

{{{graph( 330, 330, -6.5, 12.5, -5.5, 5.5,
          (x-5)/(3x)
)}}}


<B>Figure</B>. Plot y = {{{(x-5)/(3x)}}}

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