Question

How do you solve `1/(x+3) + 1/(x-3) = 1/(x^2-9)`?

Answer 1

The only configuration that yields a logical answer is:
`1/(x+3)+1/(x-3) = 1/(x^2-9)`

In which case`" "x=1/2`

Explanation:

Considering different configuration:

`color(blue)("Configuration 1")`
Suppose the Left hand side was meant to be `" "1/(x+3)+1/(x-3)`

Then the left would be:

`((x+3)+(x-3))/(x^2-9)`

Comparing left to right gives

`x+3+x-3=1`

`2x=1`

`x=1/2`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Configuration 2")`

Suppose the Left hand side was meant to be `" "1/(x+3)-1/(x-3)`

Then the left would be:

`((x+3)-(x-3))/(x^2-9)= 6/(x^2-9)`

Comparing Left to right would mean that it would have to be true for

`6=1" "color(red)("Clearly this is a contradiction so it is not the case")`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(green)("The only possible scenario is for configuration 1")`

`color(magenta)("So the answer is "x=1/2)`