Question 978999


1.  Let us assume that the denominator  x+3  is greater than  0:


x + 3 > 0,     i.e.   x > -3.


Then the inequality &nbsp;{{{(3x-4)/(x+3)}}} < {{{2}}}&nbsp; is equivalent to the inequality 

{{{3x-4}}} < {{{2}}}{{{(x+3)}}}. 


(we simply multiplied both sides of the original inequality by the positive multiplier - denominator, and kept the sign of the inequality &nbsp;"as is").


Solve it step by step:


3x - 4 < 2x + 6,

x < 6 + 4 = 10.


Thus &nbsp;-3 < x < 10&nbsp; is the solution of the given inequality.



2. &nbsp;Next, assume that the denominator &nbsp;x+3&nbsp; is lesser than &nbsp;0:


x + 3 < 0, &nbsp;&nbsp;&nbsp;&nbsp;i.e. &nbsp;&nbsp;x < -3.


Then the inequality &nbsp;{{{(3x-4)/(x+3)}}} < {{{2}}}&nbsp; is equivalent to the inequality 

{{{3x-4}}} > {{{2}}}{{{(x+3)}}}. 


(we simply multiplied both sides of the original inequality by the negative multiplier - denominator, &nbsp;and accordingly changed the sign of the inequality to the opposite one).


Solve it step by step:


3x - 4 > 2x + 6,

x > 6 + 4 = 10.


Thus this alternative lead us to the two two inequalities x < -3  and x > 10. &nbsp;Obviously, they can not be true simultaneously. &nbsp;So, &nbsp;this alternative has no solution.


The only solution is the first alternative solved in n.1.


<B>Answer</B>. The solution of the given inequality is &nbsp;-3 < x < 10.