https://file.aiquickdraw.com/mathbot/file/1729325174260dm74mu39.jpg

The inequality in the image is:

\[
\frac{x^2 - 5x - 6}{x + 2} > 0
\]

### Steps to solve this inequality:

1. **Factor the numerator**:
   Factor \(x^2 - 5x - 6\) to find its roots.
   
   \[
   x^2 - 5x - 6 = (x - 6)(x + 1)
   \]

2. **Identify critical points**:
   Set both the numerator and the denominator equal to zero to find critical points.

   - From \( (x - 6)(x + 1) = 0 \), we get \(x = 6\) and \(x = -1\).
   - From the denominator \( x + 2 = 0 \), we get \(x = -2\).

   So, the critical points are \(x = -2\), \(x = -1\), and \(x = 6\).

3. **Analyze intervals**:
   These critical points divide the real line into four intervals:
   
   - \( (-\infty, -2) \)
   - \( (-2, -1) \)
   - \( (-1, 6) \)
   - \( (6, \infty) \)
   
   We now test each interval by picking test points within them to determine whether the expression is positive or negative.

   - For \(x \in (-\infty, -2)\), say \(x = -3\):
     \[
     \frac{(x - 6)(x + 1)}{x + 2} = \frac{(-3 - 6)(-3 + 1)}{-3 + 2} = \frac{(-9)(-2)}{-1} = \frac{18}{-1} = -18 \quad (\text{negative})
     \]

   - For \(x \in (-2, -1)\), say \(x = -1.5\):
     \[
     \frac{(x - 6)(x + 1)}{x + 2} = \frac{(-1.5 - 6)(-1.5 + 1)}{-1.5 + 2} = \frac{(-7.5)(-0.5)}{0.5} = \frac{3.75}{0.5} = 7.5 \quad (\text{positive})
     \]

   - For \(x \in (-1, 6)\), say \(x = 0\):
     \[
     \frac{(x - 6)(x + 1)}{x + 2} = \frac{(0 - 6)(0 + 1)}{0 + 2} = \frac{(-6)(1)}{2} = \frac{-6}{2} = -3 \quad (\text{negative})
     \]

   - For \(x \in (6, \infty)\), say \(x = 7\):
     \[
     \frac{(x - 6)(x + 1)}{x + 2} = \frac{(7 - 6)(7 + 1)}{7 + 2} = \frac{(1)(8)}{9} = \frac{8}{9} \quad (\text{positive})
     \]

4. **Determine where the expression is positive**:
   Based on the test points, the expression is positive in the intervals \( (-2, -1) \) and \( (6, \infty) \).

5. **Conclusion**:
   The solution to the inequality is:

   \[
   (-2, -1) \cup (6, \infty)
   \]

   Note that \(x = -2\) is excluded since it makes the denominator zero, which is undefined.

Would you like more details or have any further questions? Here are some related problems:

1. What is the solution to the inequality \(\frac{x^2 - 4}{x - 3} \geq 0\)?
2. How do you solve \(\frac{(x - 1)(x + 2)}{x + 4} < 0\)?
3. How do you solve quadratic inequalities using a number line?
4. What happens when the inequality has more than two factors in the numerator?
5. How does the degree of the polynomial affect the sign chart?

**Tip**: Always remember to check for points that make the denominator zero since they define vertical asymptotes where the function is undefined!

Solve the inequality: (x^2 - 5x - 6) / (x + 2) > 0.

Learn how to solve the inequality (x^2 - 5x - 6) / (x + 2) > 0 with a step-by-step solution. This problem covers algebraic concepts such as factoring quadratic equations and using sign charts for rational expressions. Suitable for students in Grades 9-11.

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