To solve the equation \(2^{x+2} + 2^x + 2^{x-1} = \frac{11}{2}\), we can simplify it step by step.

### Step-by-Step Solution

1. **Rewrite each term in terms of \(2^x\):**

   \[
   2^{x+2} = 2^x \cdot 2^2 = 4 \cdot 2^x
   \]
   \[
   2^{x-1} = \frac{2^x}{2}
   \]

   Substitute these into the original equation:

   \[
   4 \cdot 2^x + 2^x + \frac{2^x}{2} = \frac{11}{2}
   \]

2. **Combine all the terms involving \(2^x\):**

   \[
   4 \cdot 2^x + 2^x + \frac{2^x}{2} = \frac{11}{2}
   \]

   Factor out \(2^x\):

   \[
   2^x \left(4 + 1 + \frac{1}{2}\right) = \frac{11}{2}
   \]

3. **Simplify the expression inside the parentheses:**

   \[
   4 + 1 + \frac{1}{2} = 5 + \frac{1}{2} = \frac{10}{2} + \frac{1}{2} = \frac{11}{2}
   \]

   So, the equation becomes:

   \[
   2^x \cdot \frac{11}{2} = \frac{11}{2}
   \]

4. **Divide both sides by \(\frac{11}{2}\):**

   \[
   2^x = 1
   \]

5. **Solve for \(x\):**

   Since \(2^0 = 1\), we have:

   \[
   x = 0
   \]

### Final Answer

The solution to the equation \(2^{x+2} + 2^x + 2^{x-1} = \frac{11}{2}\) is \(\boxed{x = 0}\).

Would you like more details on any part of the solution? Here are some related questions you might find interesting:

1. What happens if the coefficients of the terms in the equation are different?
2. How would the solution change if we replaced \(2^x\) with \(3^x\) in the original equation?
3. What are the general steps for solving exponential equations?
4. Can you solve \(2^{x+2} + 2^x = 6\)?
5. How would you solve an equation like \(2^{2x} + 2^{x} = 8\)?

**Tip:** When solving exponential equations, always look for ways to express all terms with the same base to simplify the equation.

Learn how to solve the exponential equation 2^(x+2) + 2^x + 2^(x-1) = 11/2 with a detailed step-by-step solution. This problem involves exponential equations and requires understanding of exponent rules and simplification techniques. Suitable for students in Grades 10-12.

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