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The problem asks to determine the set of solutions for the following inequality on the interval \( ] -3, +\infty [ \):

\[
\ln(x + 5) + \ln(x + 3) \geq 2 \ln(x + 3)
\]

### Step 1: Simplify the inequality

We start by simplifying the inequality. First, we combine the logarithmic terms on the left-hand side:

\[
\ln(x + 5) + \ln(x + 3) = \ln((x + 5)(x + 3))
\]

The inequality becomes:

\[
\ln((x + 5)(x + 3)) \geq 2 \ln(x + 3)
\]

Next, use the logarithmic property \( a \ln(b) = \ln(b^a) \) to rewrite the right-hand side:

\[
\ln((x + 5)(x + 3)) \geq \ln((x + 3)^2)
\]

### Step 2: Eliminate the logarithms

Since the logarithmic functions are defined for positive arguments and they are increasing functions, we can drop the logarithms as long as the arguments are positive. Therefore, the inequality becomes:

\[
(x + 5)(x + 3) \geq (x + 3)^2
\]

### Step 3: Expand both sides

Expand both sides of the inequality:

\[
(x + 5)(x + 3) = x^2 + 8x + 15
\]
\[
(x + 3)^2 = x^2 + 6x + 9
\]

Now substitute these into the inequality:

\[
x^2 + 8x + 15 \geq x^2 + 6x + 9
\]

### Step 4: Simplify the inequality

Subtract \( x^2 \) from both sides:

\[
8x + 15 \geq 6x + 9
\]

Subtract \( 6x \) and \( 9 \) from both sides:

\[
2x \geq -6
\]

Finally, divide by 2:

\[
x \geq -3
\]

### Step 5: Determine the domain

We need to check if this solution satisfies the domain restriction given in the problem, which is \( ] -3, +\infty [ \). The solution \( x \geq -3 \) suggests that \( x = -3 \) could be a boundary point, but it is not included in the interval due to the open interval notation \( ]-3, +\infty[ \).

So the solution set is:

\[
x \in ] -3, +\infty [
\]

### Final Answer

The solution set is:

\[
] -3, +\infty [
\]

Would you like more details or have any questions?

### Related Questions:
1. How would the solution change if the inequality were strict (\( > \)) instead of \( \geq \)?
2. What happens if we change the domain to \( ]-5, +\infty[ \)?
3. How does the logarithmic inequality behave for different base values of logarithms (other than natural logarithms)?
4. Can the same method be applied to inequalities involving other types of functions, such as exponentials?
5. What is the geometric interpretation of the solution set for this inequality?

**Tip:** Always check the domain of logarithmic functions before solving an inequality to avoid undefined values.

Déterminer l'ensemble des solutions sur ]-3; +∞[ de: ln(x + 5) + ln(x + 3) ≥ 2 ln(x + 3)

Learn how to solve the logarithmic inequality ln(x + 5) + ln(x + 3) ≥ 2 ln(x + 3) within the interval ]-3, +∞[. This problem demonstrates logarithmic properties, domain restrictions, and algebraic simplifications, suitable for students in Grades 10-12.

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