SOLUTION: 3^[(x+1)/(x-5)] + 3^[(3x-9)/(x-5)] = 10/3. Find x.

Algebra.Com
Question 1208229: 3^[(x+1)/(x-5)] + 3^[(3x-9)/(x-5)] = 10/3.
Find x.
Found 2 solutions by mccravyedwin, ikleyn:
Answer by mccravyedwin(405)   (Show Source): You can put this solution on YOUR website!


I noticed that 




Then it hit me! We have the sum of two powers of 3 on the left, and
the sum of two powers of 3 on the right.  So I asked myself: "Do you
suppose the two exponents on the left could equal to the two exponents 
on the right? It's worth a try to see.

So I tried seeing if the first exponent on the left could be equal to
the first exponent on the right.





So that's no good.  But then I tried setting the first exponent on the 
left equal to the second exponent on the right.






So sure enough! So we've got it solved! because the other exponents now
have to be equal.

Although I know they have to be equal and the answer has to be 2,
I'll show it anyway:






Answer:  x = 2

Edwin
Answer by ikleyn(52765)   (Show Source): You can put this solution on YOUR website!
.
.
3^[(x+1)/(x-5)] + 3^[(3x-9)/(x-5)] = 10/3.
Find x.
~~~~~~~~~~~~~~~~~~


        I will solve it informally.  It means that I will use
        my common sense in order for it lit my way in the darkness. 


Right side  10/3  is   = 3 = 3 + .


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


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

      = 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  .  It leads me to this equation for exponent 

     = -1  -->  x+1 = -x + 5  -->  2x = 4  -->  x = 4/2 = 2.



With x= 2, the exponent in the second addend is   =  =  = 1.


It is exactly what I need.


So,     is the solution.    ANSWER

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;  print another equation  y = 10/3  for the right side function.

The plot shows that the found solution  x= 2  is  .




RELATED QUESTIONS

-(3x+5)=9(x+3) (answered by stanbon)
1.) 3x+12=15/x 2.)... (answered by josgarithmetic)
1.) 3x+12=15/x 2.)... (answered by kekegeter )
Divide.... (answered by algebrahouse.com)
3x-5+3(x+3)=10 (answered by rfer)
3(x+2)-2(3x-5)=10 find... (answered by jim_thompson5910)
having trouble with these problems. 3(2-4x)+9(x-5)-10(3-2x)-7(X-3)=5... (answered by Edwin McCravy)
3x-5=x+3 (answered by checkley75)
3x-5=x+3 (answered by harpazo)