To solve the inequality

\[
\frac{5x + 30}{x^2 + 10x + 24} + 1 < 0,
\]

we'll follow similar steps as before but consider when the entire expression is negative.

### Step 1: Combine the Terms into a Single Fraction

Rewrite the inequality by combining the terms into a single fraction:

\[
\frac{5x + 30}{x^2 + 10x + 24} + 1 = \frac{5x + 30 + (x^2 + 10x + 24)}{x^2 + 10x + 24}.
\]

Simplify the numerator:

\[
5x + 30 + x^2 + 10x + 24 = x^2 + 15x + 54.
\]

Thus, the inequality becomes:

\[
\frac{x^2 + 15x + 54}{x^2 + 10x + 24} < 0.
\]

### Step 2: Factor the Numerator and Denominator

Now, we factor both the numerator and the denominator:

- Numerator: \(x^2 + 15x + 54 = (x + 6)(x + 9)\)
- Denominator: \(x^2 + 10x + 24 = (x + 4)(x + 6)\)

Substitute these factorizations back into the inequality:

\[
\frac{(x + 6)(x + 9)}{(x + 4)(x + 6)} < 0.
\]

### Step 3: Determine the Critical Points

The critical points occur where the numerator and the denominator are zero:

- \(x + 6 = 0 \implies x = -6\)
- \(x + 9 = 0 \implies x = -9\)
- \(x + 4 = 0 \implies x = -4\)

### Step 4: Analyze the Sign of the Expression in Each Interval

The critical points divide the number line into four intervals:

1. \(x < -9\)
2. \(-9 < x < -6\)
3. \(-6 < x < -4\)
4. \(x > -4\)

We need to test each interval to determine where the expression \(\frac{(x + 6)(x + 9)}{(x + 4)(x + 6)}\) is negative.

#### Interval Testing

1. **For \(x < -9\) (e.g., \(x = -10\)):**

   - \(x + 9 < 0\)
   - \(x + 6 < 0\)
   - \(x + 4 < 0\)

   All terms are negative, so the expression is positive.

2. **For \(-9 < x < -6\) (e.g., \(x = -8\)):**

   - \(x + 9 > 0\)
   - \(x + 6 < 0\)
   - \(x + 4 < 0\)

   The numerator is negative, and the denominator is positive, so the expression is negative.

3. **For \(-6 < x < -4\) (e.g., \(x = -5\)):**

   - \(x + 9 > 0\)
   - \(x + 6 > 0\)
   - \(x + 4 < 0\)

   The numerator is positive, and the denominator is negative, so the expression is negative.

4. **For \(x > -4\) (e.g., \(x = 0\)):**

   - \(x + 9 > 0\)
   - \(x + 6 > 0\)
   - \(x + 4 > 0\)

   All terms are positive, so the expression is positive.

### Step 5: Conclusion

The expression \(\frac{(x + 6)(x + 9)}{(x + 4)(x + 6)} < 0\) is true for the intervals:

\[
-9 < x < -6 \quad \text{and} \quad -6 < x < -4.
\]

However, since \(x = -6\) makes the denominator zero, it is excluded from the solution.

Thus, the solution set for the inequality is:

\[
(-9, -6) \cup (-6, -4).
\]

Do you want more details or have any questions?

Here are 5 related questions to expand on this topic:

1. How do you solve inequalities involving rational expressions?
2. What happens when a critical point makes the denominator zero in inequalities?
3. How do you test intervals for the sign of a rational expression?
4. Can you explain why a rational expression is undefined at certain points?
5. What are some common mistakes to avoid when solving inequalities?

**Tip:** Always exclude points from the solution set where the denominator is zero, as these are not within the domain of the rational expression.

Learn how to solve the inequality (5x + 30) / (x^2 + 10x + 24) + 1 < 0 with a detailed step-by-step solution. This problem covers key concepts in inequalities and rational expressions, including critical points and interval testing. Suitable for students in Grades 11-12.

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