solve the following rational inequality: (x^2+2x-8)/(x+2)>0

We are tasked with solving the rational inequality:

\[
\frac{x^2 + 2x - 8}{x + 2} > 0
\]

### Step 1: Factor the numerator and denominator
Start by factoring the quadratic expression in the numerator.

\[
x^2 + 2x - 8 = (x - 2)(x + 4)
\]

Thus, the inequality becomes:

\[
\frac{(x - 2)(x + 4)}{x + 2} > 0
\]

### Step 2: Determine the critical points
The critical points occur where the expression is equal to zero or undefined (where the denominator is zero). These occur when:

- \( x - 2 = 0 \Rightarrow x = 2 \)
- \( x + 4 = 0 \Rightarrow x = -4 \)
- \( x + 2 = 0 \Rightarrow x = -2 \)

Thus, the critical points are \( x = -4, -2, 2 \).

### Step 3: Analyze the sign of the expression in intervals
To determine the solution to the inequality, we analyze the sign of the expression on the intervals defined by the critical points: \( (-\infty, -4) \), \( (-4, -2) \), \( (-2, 2) \), and \( (2, \infty) \).

We'll pick test points in each interval and evaluate the sign of the expression:

- **Interval \( (-\infty, -4) \)**: Pick \( x = -5 \)
  - \( \frac{(x - 2)(x + 4)}{x + 2} = \frac{(-5 - 2)(-5 + 4)}{-5 + 2} = \frac{(-7)(-1)}{-3} < 0 \)
- **Interval \( (-4, -2) \)**: Pick \( x = -3 \)
  - \( \frac{(x - 2)(x + 4)}{x + 2} = \frac{(-3 - 2)(-3 + 4)}{-3 + 2} = \frac{(-5)(1)}{-1} > 0 \)
- **Interval \( (-2, 2) \)**: Pick \( x = 0 \)
  - \( \frac{(x - 2)(x + 4)}{x + 2} = \frac{(0 - 2)(0 + 4)}{0 + 2} = \frac{(-2)(4)}{2} < 0 \)
- **Interval \( (2, \infty) \)**: Pick \( x = 3 \)
  - \( \frac{(x - 2)(x + 4)}{x + 2} = \frac{(3 - 2)(3 + 4)}{3 + 2} = \frac{(1)(7)}{5} > 0 \)

### Step 4: Identify the solution intervals
We need the expression to be positive (\( > 0 \)), so we choose the intervals where the sign is positive:

- \( (-4, -2) \)
- \( (2, \infty) \)

### Step 5: Check the critical points
At \( x = -4 \) and \( x = 2 \), the numerator is zero, so the expression equals zero at these points, and they are not included in the solution. At \( x = -2 \), the expression is undefined, so \( x = -2 \) is also not included.

### Final Solution
The solution to the inequality is:

\[
x \in (-4, -2) \cup (2, \infty)
\]

Would you like more details on any step or have any questions?

Here are 5 related questions:
1. How do we handle rational inequalities with higher-degree polynomials?
2. What happens when a rational inequality is less than or equal to zero?
3. How do vertical asymptotes affect the solution of rational inequalities?
4. How do you check if a critical point is included in the solution?
5. How can rational inequalities be applied in real-life problems?

**Tip:** Always test points from each interval when solving rational inequalities to determine the correct sign!

This guide explains how to solve the rational inequality (x^2 + 2x - 8) / (x + 2) > 0 step-by-step. It includes factoring quadratics, finding critical points, and performing sign analysis. Ideal for students in Grades 9-11.

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