To solve the equation 3^x:2 + 3^x - 2 = 2^3.11, we can simplify it step by step.

First, let's simplify the left side of the equation:
3^x:2 + 3^x - 2

Since 3^x:2 means 3^x divided by 2, we can rewrite the equation as:
(3^x)/2 + 3^x - 2

Now, let's simplify the right side of the equation:
2^3.11

To simplify this, we can rewrite it as:
2^(3 + 0.11)

Using the property of exponents, we can rewrite it as:
2^3 * 2^0.11

2^3 is equal to 8, so we have:
8 * 2^0.11

Now, let's go back to the left side of the equation:
(3^x)/2 + 3^x - 2

To combine the terms, we need a common denominator. The common denominator for 2 and 1 is 2, so we can rewrite the equation as:
(3^x)/2 + (2*3^x)/2 - 2

Now, we can combine the terms:
(3^x + 2*3^x - 4)/2

To simplify further, we can factor out 3^x:
(3^x(1 + 2) - 4)/2

Simplifying the numerator:
(3^x(3) - 4)/2

Now, we can rewrite the equation as:
(3*3^x - 4)/2 = 8 * 2^0.11

Multiplying both sides of the equation by 2 to eliminate the denominator:
3*3^x - 4 = 16 * 2^0.11

Now, let's simplify the right side of the equation:
16 * 2^0.11

Using a calculator, we find that 2^0.11 is approximately 1.0798. Multiplying 16 by 1.0798, we get:
16 * 1.0798 ≈ 17.2768

So, the equation becomes:
3*3^x - 4 = 17.2768

Adding 4 to both sides of the equation:
3*3^x = 21.2768

Dividing both sides of the equation by 3:
3^x = 21.2768/3

Using a calculator, we find that 21.2768/3 is approximately 7.0923.

So, the equation becomes:
3^x = 7.0923

To solve for x, we need to take the logarithm of both sides of the equation. Let's take the logarithm base 3:
log3(3^x) = log3(7.0923)

Using the property of logarithms, we can rewrite the equation as:
x = log3(7.0923)

Using a calculator, we find that log3(7.0923) is approximately 1.6831.

Therefore, the solution to the equation 3^x:2 + 3^x - 2 = 2^3.11 is x ≈ 1.6831.