To solve this equation, we need to isolate the variable x.

First, let's get rid of the fractions by multiplying every term by the least common denominator, which is 4x:

4x(x + 1/4) = 4x(9/x + 1)

Expanding both sides:

4x^2 + x = 36 + 4x

Next, let's move all terms to one side of the equation:

4x^2 + x - 4x - 36 = 0

Combining like terms:

4x^2 - 3x - 36 = 0

Now, we can solve this quadratic equation. Factoring may not work easily, so let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 4, b = -3, and c = -36. Plugging these values into the quadratic formula:

x = (-(-3) ± √((-3)^2 - 4(4)(-36))) / (2(4))

Simplifying:

x = (3 ± √(9 + 576)) / 8

x = (3 ± √585) / 8

Therefore, the solutions to the equation are:

x = (3 + √585) / 8

x = (3 - √585) / 8