Math Problem Statement

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

Solution

The equation in the image asks to solve the inequality:

ex5e5x+3e^{-x-5} \leq e^{-5x+3}

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

Step 1: Take the natural logarithm of both sides

Since the exponential function exe^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:

x55x+3-x - 5 \leq -5x + 3

Step 2: Simplify the inequality

Now, we simplify this linear inequality:

x55x+3-x - 5 \leq -5x + 3

First, move all terms involving xx to one side and constants to the other side:

x+5x3+5- x + 5x \leq 3 + 5

4x84x \leq 8

Step 3: Solve for xx

Divide both sides by 4:

x2x \leq 2

Step 4: Conclusion

The solution is the set of real numbers xx such that x2x \leq 2. Therefore, the solution set in interval notation is:

(,2](-\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 ee?
  3. How does the strict monotonicity of exe^x help in solving inequalities?
  4. What are the steps to graph ex5e^{-x-5} and e5x+3e^{-5x+3}?
  5. Can you derive a similar solution for a different base like 2x2^x?

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

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Math Problem Analysis

Mathematical Concepts

Algebra
Exponential Inequalities
Logarithmic Properties

Formulas

ln(e^a) = a
Exponential Inequality: e^x ≤ e^y implies x ≤ y

Theorems

Properties of Exponents
Logarithmic Properties
Monotonicity of Exponential Functions

Suitable Grade Level

Grades 10-12