Question 1194429
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The numerator of the fraction is x, while the denominator is 5 more the numerator.
if the value of the fraction is less than 2/5, find the value of x
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<pre>
Let x be the numerator;  then the denominator is (x+5) and the fraction is  {{{x/(x+5)}}}.

They want you solve this inequality

    {{{x/(x+5)}}} < {{{2/5}}}.     (1)


                        +---------------------+
                        |     Attention (!)   |
                        +---------------------+


     Do not multiply both sides by (x+5), because you don't know in advance, 
                  if this value is positive or negative.


    +----------------------------------------------------------------------+
    |   THERE IS ANOTHER METHOD TO SOLVE IT without making wrong steps !   |
    +----------------------------------------------------------------------+


Re-write equation (1)  <U>equivalently</U>  in this form

    {{{x/(x+5)}}} - {{{2/5}}} < 0.


Write it with the common denominator, which is  5*(x+5).

    {{{(5x)/(5*(x+5))}}} - {{{(2*(x+5))/(5*(x+5))}}} < 0.


Simplify

    {{{(5x - 2*(x+5))/(5*(x+5))}}} < 0,

    {{{(5x - 2x - 10)/(5*(x+5))}}} < 0,

    {{{(3x - 10)/(5(x+5))}}} < 0.


This inequality is equivalent to

    {{{(3x-10)/(x+5)}}} < 0.


The last inequality is hold in these two cases:

    (a)  3x-10 < 0  and  x+5 > 0

or 

    (b)  3x-10 > 0  and  x+5 < 0.



In case (a),  it implies  3x < 10  and  x > -5,  which is the same as  x < 10/3  and  x > -5,

              which gives the answer -5 < x < 10/3.


In case (b),  it implies  3x > 10  and  x < -5,  which  has no solutions.


<U>ANSWER</U>.  The solution set is  -5 < x < 10/3,  or,  in interval form,  (-5, 10/3).
</pre>

Solved.