Solving Exponential Inequality: e^((-x/2)

Math Problem Statement

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

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

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

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:

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Exponential Functions
Inequalities
Logarithms

e^a ≤ e^b implies a ≤ b
Natural logarithm ln(e^x) = x

Properties of Exponential and Logarithmic Functions

Grades 10-12

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Solving Exponential Equation: e^(-2x - 5) = e^x

Solving Exponential and Logarithmic Inequalities