https://file.aiquickdraw.com/mathbot/file/17270283630044fnoojsl.jpg

The inequality in the image is:

\[
\frac{-4}{x+2} + \frac{7}{x+3} > 0
\]

### Solution:

1. **Find the common denominator**: The denominators are \(x+2\) and \(x+3\), so the common denominator is \((x+2)(x+3)\).

\[
\frac{-4}{x+2} + \frac{7}{x+3} = \frac{-4(x+3)}{(x+2)(x+3)} + \frac{7(x+2)}{(x+2)(x+3)}
\]

2. **Simplify the numerators**:

\[
\frac{-4(x+3) + 7(x+2)}{(x+2)(x+3)} = \frac{-4x - 12 + 7x + 14}{(x+2)(x+3)}
\]

\[
= \frac{3x + 2}{(x+2)(x+3)}
\]

3. **Solve the inequality**:

We now need to solve:

\[
\frac{3x + 2}{(x+2)(x+3)} > 0
\]

The sign of the expression depends on the sign of the numerator and denominator. Let's analyze the critical points:

- **Numerator**: \(3x + 2 = 0\) gives \(x = -\frac{2}{3}\).
- **Denominator**: The denominator is zero at \(x = -2\) and \(x = -3\).

So, the critical points are \(x = -3\), \(x = -2\), and \(x = -\frac{2}{3}\).

### Sign Analysis:

We divide the number line into intervals based on these critical points: \((- \infty, -3)\), \((-3, -2)\), \((-2, -\frac{2}{3})\), and \((- \frac{2}{3}, \infty)\).

1. **For \(x \in (-\infty, -3)\)**:
   - \(3x + 2 < 0\)
   - \((x+2) < 0\)
   - \((x+3) < 0\)
   - The overall sign is negative.

2. **For \(x \in (-3, -2)\)**:
   - \(3x + 2 < 0\)
   - \((x+2) < 0\)
   - \((x+3) > 0\)
   - The overall sign is positive.

3. **For \(x \in (-2, -\frac{2}{3})\)**:
   - \(3x + 2 > 0\)
   - \((x+2) > 0\)
   - \((x+3) > 0\)
   - The overall sign is positive.

4. **For \(x \in (-\frac{2}{3}, \infty)\)**:
   - \(3x + 2 > 0\)
   - \((x+2) > 0\)
   - \((x+3) > 0\)
   - The overall sign is positive.

### Solution:

- The inequality is positive in the intervals \((-3, -2)\) and \((-2, \infty)\).

### Final Answer:

\[
x \in (-3, -2) \cup \left( -\frac{2}{3}, \infty \right)
\]

Let me know if you'd like further details or clarifications.

---

Here are 5 related questions:
1. How would the solution change if the inequality were \( \leq 0 \)?
2. How do you graph the solution of this inequality on a number line?
3. What is the importance of critical points in solving rational inequalities?
4. Can this inequality be solved using interval notation directly without factoring?
5. How does the sign of a rational expression change at its critical points?

**Tip:** Always verify the points where the denominator becomes zero, as the expression is undefined at those values.

Solve the inequality: -4/(x+2) + 7/(x+3) > 0

Learn how to solve the rational inequality -4/(x+2) + 7/(x+3) > 0 with a detailed step-by-step solution. This problem covers key algebraic concepts such as rational expressions, common denominator methods, and sign analysis. Suitable for students in Grades 9-12.

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