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The problem presented is to determine the set of real solutions \( \mathbb{R} \) for the inequality:

\[
e^{\frac{-x}{2} - 5} \leq e^{-5x + 3}
\]

### Steps to solve:

1. **Simplify the inequality:**
   Since both sides are exponentials with the same base \( e \), we can take the natural logarithm (ln) on both sides, which simplifies the inequality as follows:

   \

Déterminer l'ensemble des solutions sur ℝ de : e^((-x/2) - 5) ≤ e^(-5x + 3)

Learn how to solve the exponential inequality e^((-x/2) - 5) ≤ e^(-5x + 3) with step-by-step guidance. This example explores key concepts such as exponential functions, inequalities, and logarithmic properties. Suitable for high school students in Grades 10-12.

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