exp equation

i am just writing a soln.. but don't know weather its right or wrong..

clearly, 2^1+x + 2^1-x + 3^1+x + 3^1-x are all positive and real.. for real x .. so why not A.M-G.M

so, we get, 5^1+x + 5^1-x > (36)^1/4

or, 5^x + 5^-x > 6^1/2/5

or, 5^x > 6^1/2/5 - 5^-x

=> x > log₅(6^1/2/5 - 5^-x)

comparison can be made only if the terms are real...

and for R.H.S to be real...
the definiton of log should be used...

i.e, 5^-x < 6^1/2/5

again, taking logs...
i know its too much but there's no harm if are confirmed that the terms are positive and also the bases i.e. greater than 0 less than 1 or greater than 1...

=> -x < log₅6^1/2 - 1

=> x > 1 - log₅6^1/2

this is the x for which we got x to be defined for...

but i don't think that this the answer as well...