To solve this inequality, we will first simplify each term:

Starting with the left-hand side:

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

= [(3^x + 9)^2 + (3^x - 9)^2] / [(3^x - 9)(3^x + 9)]

= [2(3^(2x) + 81)] / [(3^x - 9)(3^x + 9)]

Next, simplifying the right-hand side:

4*((3^(x+1))+144)/((9^x)-81)

= 4*(3*3^x + 144) / (9^x - 81)

= (12*3^x + 576) / (9^x - 81)

Now, we can substitute u = 3^x and simplify further:

[2(u^2 + 81)] / [(u - 3)(u + 3)] >= (12u + 576) / (u^2 - 81)

Multiplying both sides by (u^2 - 81) * (u - 3) * (u + 3), we get:

2(u^2 + 81)(u^2 - 9) >= (12u + 576)(u - 3)(u + 3)

Expanding both sides and simplifying, we obtain:

2u^4 - 192u^2 - 864u - 3888 >= 0

Dividing both sides by 2 and factoring, we get:

(u - 6)(u + 6)(u^2 - 72) >= 0

Since u = 3^x, we can write this as:

(3^x - 6)(3^x + 6)(3^x - 3v8)(3^x + 3v8) >= 0

The roots of this inequality are -3v8, -6, 3v8, and 6. We can use test points to determine the sign of the inequality between these roots:

When x < log3(3v8), all four factors are negative, so the inequality is satisfied.

When log3(3v8) < x < log3(6), the first three factors are negative and the last factor is positive, so the inequality is not satisfied.

When x > log3(6), all four factors are positive, so the inequality is satisfied.

Therefore, the solution to the inequality is:

x ? (-?, log3(3v8)) U (log3(6), ?)