https://file.aiquickdraw.com/mathbot/file/1726678867576gbnmd3p5.jpg

The equation in the image asks to solve the inequality:

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

We need to determine the set of solutions in \( \mathbb{R} \). Let's solve it step by step.

### Step 1: Take the natural logarithm of both sides
Since the exponential function \( e^x \) is strictly increasing, the inequality holds true if and only if the exponents satisfy the same inequality. So we can take the natural logarithm on both sides without changing the direction of the inequality:

\[
-x - 5 \leq -5x + 3
\]

### Step 2: Simplify the inequality
Now, we simplify this linear inequality:

\[
-x - 5 \leq -5x + 3
\]

First, move all terms involving \( x \) to one side and constants to the other side:

\[
- x + 5x \leq 3 + 5
\]

\[
4x \leq 8
\]

### Step 3: Solve for \( x \)
Divide both sides by 4:

\[
x \leq 2
\]

### Step 4: Conclusion
The solution is the set of real numbers \( x \) such that \( x \leq 2 \). Therefore, the solution set in interval notation is:

\[
(-\infty, 2]
\]

Would you like further clarification or details on this solution? Here are some related questions to explore:

1. How do we apply logarithmic properties to inequalities?
2. What happens if the base of the exponent is not \( e \)?
3. How does the strict monotonicity of \( e^x \) help in solving inequalities?
4. What are the steps to graph \( e^{-x-5} \) and \( e^{-5x+3} \)?
5. Can you derive a similar solution for a different base like \( 2^x \)?

**Tip**: Always remember to check whether the inequality direction changes when dividing or multiplying by negative numbers!

Déterminer l'ensemble des solutions sur ℝ de : e^{-x-5} ≤ e^{-5x+3}

Learn how to solve the inequality e^{-x-5} ≤ e^{-5x+3}. This solution applies logarithmic properties and the behavior of exponential functions. Suitable for high school students in Grades 10-12.

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