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The given equation is:

\[
e^{2x - 2} = e^{-x - 3}
\]

We need to determine the set of solutions in \(\mathbb{R}\).

### Step-by-Step Solution:

1. Since both sides of the equation have the same base \(e\), we can set the exponents equal to each other:
   \[
   2x - 2 = -x - 3
   \]

2. Solve for \(x\):
   \[
   2x + x = -3 + 2
   \]
   \[
   3x = -1
   \]
   \[
   x = -\frac{1}{3}
   \]

So, the solution to the equation is \(x = -\frac{1}{3}\).

### Final Answer:
The solution set is \(\left\{-\frac{1}{3}\right\}\).

Would you like more details or have any questions?

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Here are 5 related questions to further explore this concept:
1. How do we solve exponential equations where the base is not \(e\)?
2. What happens if the equation involves different bases?
3. Can this equation have multiple solutions if the exponents were quadratic?
4. What is the relationship between logarithms and exponential equations?
5. How do we solve similar equations if complex numbers are involved?

**Tip:** For equations with the same base, always equate the exponents directly to simplify the solving process.

Déterminer l'ensemble des solutions dans ℝ de: e^(2x-2) = e^(-x-3)

Learn how to solve the exponential equation e^(2x - 2) = e^(-x - 3) with a detailed step-by-step solution. This problem covers key concepts in algebra and exponentials, focusing on equating exponents when bases are the same. Suitable for students in Grades 9-12.

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