To prove the inequality, we can start by simplifying the left side of the equation:

a/b + 1 + b/a + 1 + 1/(a+b) >= 3/2

Multiplying each term by (a+b), we get:

(a+b)(a/b + 1) + (a+b)(b/a + 1) + (a+b)(1/(a+b)) >= 3/2(a+b)

Simplifying further:

a^2/b + a + b^2/a + b + 1 >= (3/2)(a+b)

Now, let's focus on the left side of the equation:

a^2/b + a + b^2/a + b + 1

To simplify this expression, we can find a common denominator for the first two terms and the next two terms:

(a^2/a + ab/b) + (b^2/b + ab/a) + 1

Simplifying further:

a + ab/b + b + ab/a + 1

Now, let's simplify the middle term:

ab/b = a

Similarly, ab/a = b

Substituting these values back into the expression:

a + a + b + b + 1

Combining like terms:

2a + 2b + 1

Now, let's go back to the original inequality:

2a + 2b + 1 >= (3/2)(a+b)

Multiplying both sides by 2 to eliminate the fraction:

4a + 4b + 2 >= 3(a+b)

Expanding:

4a + 4b + 2 >= 3a + 3b

Subtracting 3a and 3b from both sides:

a + b + 2 >= 0

Since a and b are positive numbers, a + b + 2 will always be greater than or equal to 0. Therefore, the inequality holds true:

a/b + 1 + b/a + 1 + 1/(a+b) >= 3/2