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

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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) About Me  (Show Source):
You can put this solution on YOUR website!
matrix%282%2C1%2C%22%22%2C3%5E%28%28x%2B1%29%2F%28x-3%29%29%29%22%22%2B%22%22matrix%282%2C1%2C%22%22%2C3%5E%28%283x-9%29%2F%28x-3%29%29%29%22%22=%22%2210%2F3

I noticed that 10%2F3%22%22=%22%223%2B1%2F3%22%22=%22%22%22%22=%22%223%5E1%2B3%5E%28-1%29


matrix%282%2C1%2C%22%22%2C3%5E%28%28x%2B1%29%2F%28x-3%29%29%29%22%22%2B%22%22matrix%282%2C1%2C%22%22%2C3%5E%28%283x-9%29%2F%28x-3%29%29%29%22%22=%22%223%5E1%2B3%5E%28-1%29

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.

%28x%2B1%29%2F%28x-5%29=1
x%2B1=x-5
1=-5

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

%28x%2B1%29%2F%28x-5%29=-1
x%2B1=-x%2B5
2x=4
x=2

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:

%283x-9%29%2F%28x-5%29=1
3x-9=x-5
2x=4
x=2

Answer:  x = 2

Edwin

Answer by ikleyn(52748) About Me  (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  10%2F3 = 31%2F3 = 3 + 1%2F3.


It tells me that one of the addend in the left side is 3, while the other is  1%2F3.


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

     %28x%2B1%29%2F%28x-5%29 = 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%2F3.  It leads me to this equation for exponent 

    %28x%2B1%29%2F%28x-5%29 = -1  -->  x+1 = -x + 5  -->  2x = 4  -->  x = 4/2 = 2.



With x= 2, the exponent in the second addend is  %283x-9%29%2F%28x-5%29 = %283%2A2-9%29%2F%282-5%29 = %28-3%29%2F%28-3%29 = 1.


It is exactly what I need.


So,   highlight%28highlight%28x=2%29%29  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  highlight%28highlight%28UNIQUE%29%29.