Question 1208229
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3^[(x+1)/(x-5)] + 3^[(3x-9)/(x-5)] = 10/3. 
Find x. 
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        I will solve it informally.  It means that I will use 

        my common sense in order for it lit my way in the darkness.



<pre>
Right side  10/3  is  {{{10/3}}} = 3{{{1/3}}} = 3 + {{{1/3}}}.


It tells me that one of the addend in the left side is 3, while the other is  {{{1/3}}}.


OK.   I try to have first addend equal to 3.   It leads me to this equation for exponent 

     {{{(x+1)/(x-5)}}} = 1  -->  x+1 = x-5 ---> 1 = -5,   which is impossible,  so this way does not work.



OK.  I then try to have first addend equal to  {{{1/3}}}.  It leads me to this equation for exponent 

    {{{(x+1)/(x-5)}}} = -1  -->  x+1 = -x + 5  -->  2x = 4  -->  x = 4/2 = 2.



With x= 2, the exponent in the second addend is  {{{(3x-9)/(x-5)}}} = {{{(3*2-9)/(2-5)}}} = {{{(-3)/(-3)}}} = 1.


It is exactly what I need.


So,   {{{highlight(highlight(x=2))}}}  is the solution.    <U>ANSWER</U>
</pre>

Solved.


However, my solution does not guarantee that there is no another solution.


So, to check my answer, I used plotting tool DESMOS available online for free

www.desmos/calculator


You also can do it and repeat my steps.


Print equation of the function in the left side; &nbsp;print another equation &nbsp;y = 10/3 &nbsp;for the right side function.


The plot shows that the found solution &nbsp;x= 2 &nbsp;is &nbsp;{{{highlight(highlight(UNIQUE))}}}.