To solve this equation, we need to find the value of x.

First, let's simplify the equation:

1/2 * 4 + 1/4 * 6 + 1/6 * 8 + ... + 1/x * (x + 2) = 5/22

To find a pattern, let's look at the first few terms:

1/2 * 4 = 2/2 = 1
1/4 * 6 = 6/4 = 3/2
1/6 * 8 = 8/6 = 4/3

We can see that the numerator of each term is increasing by 1, and the denominator is increasing by 2. So, the nth term can be written as (n+1)/(2n).

Now, let's rewrite the equation using this pattern:

(1/2 * 4) + (1/4 * 6) + (1/6 * 8) + ... + (1/x * (x + 2)) = 5/22

(1/2 * 4) + (1/4 * 6) + (1/6 * 8) + ... + ((x+1)/(2x)) = 5/22

Now, let's find a common denominator for all the terms:

(2/2 * 4) + (3/2 * 6) + (4/3 * 8) + ... + ((x+1)/(2x)) = 5/22

(8/2) + (18/2) + (32/3) + ... + ((x+1)/(2x)) = 5/22

Now, let's combine the fractions:

(8 + 18 + 32 + ... + (x+1))/(2x) = 5/22

To find the sum of the fractions, we can use the formula for the sum of an arithmetic series:

Sum = (n/2)(first term + last term)

In this case, the first term is 8, and the last term is (x+1). The number of terms is x/2.

So, the sum of the fractions is:

((x/2)/2)(8 + (x+1)) = 5/22

Simplifying further:

(x/4)(9 + x) = 5/22

Expanding:

9x/4 + x^2/4 = 5/22

Multiplying both sides by 4 to get rid of the denominators:

9x + x^2 = 20/22

Simplifying:

x^2 + 9x - 20/22 = 0

Multiplying both sides by 22 to get rid of the fraction:

22x^2 + 198x - 20 = 0

Now, we can solve this quadratic equation. However, it seems that there might be a mistake in the equation or the given sum, as the quadratic equation does not have rational solutions.