Solving the Rational Inequality:  (x+2)/(x-3) ≥ 0

To solve the rational inequality:

\[
\frac{x+2}{x-3} \geq 0
\]

### Steps:

1. **Find critical points**: These are values of \(x\) that make the numerator or denominator equal to zero.
   - Set \(x+2 = 0 \Rightarrow x = -2\) (numerator).
   - Set \(x-3 = 0 \Rightarrow x = 3\) (denominator).

   So, the critical points are \(x = -2\) and \(x = 3\). Notice that \(x = 3\) will make the denominator zero, which means the function is undefined at this point.

2. **Create test intervals**: Divide the real line based on the critical points:
   - Interval 1: \( (-\infty, -2) \)
   - Interval 2: \( (-2, 3) \)
   - Interval 3: \( (3, \infty) \)

3. **Test the sign in each interval**: Pick a test point from each interval and determine whether the expression is positive or negative.

   - For \(x = -3\) (from interval \((-∞, -2)\)):
     \[
     \frac{-3+2}{-3-3} = \frac{-1}{-6} = \frac{1}{6} > 0
     \]
     The expression is positive in this interval.
   
   - For \(x = 0\) (from interval \((-2, 3)\)):
     \[
     \frac{0+2}{0-3} = \frac{2}{-3} = -\frac{2}{3} < 0
     \]
     The expression is negative in this interval.

   - For \(x = 4\) (from interval \((3, ∞)\)):
     \[
     \frac{4+2}{4-3} = \frac{6}{1} = 6 > 0
     \]
     The expression is positive in this interval.

4. **Analyze the critical points**:
   - At \(x = -2\), the numerator is zero, so \(\frac{x+2}{x-3} = 0\). This satisfies the inequality because we are looking for \(\geq 0\).
   - At \(x = 3\), the denominator is zero, so the expression is undefined. Hence, \(x = 3\) is excluded from the solution.

5. **Conclusion**:
   The solution includes the intervals where the expression is positive or zero:
   \[
   (-\infty, -2] \cup (3, \infty)
   \]

### Final answer:
\[
x \in (-\infty, -2] \cup (3, \infty)
\]

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Would you like more details or have any questions? 

Here are 5 related questions:
1. How do we solve rational inequalities with quadratic expressions?
2. What happens if both numerator and denominator contain factors?
3. Can we solve rational inequalities using a sign chart method?
4. How do we deal with strict inequalities (e.g., \(> 0\)) in rational functions?
5. How do we interpret the graph of a rational function with inequality constraints?

**Tip:** Always check critical points where the expression is undefined by setting the denominator to zero!

Solving the Rational Inequality: (x+2)/(x-3) ≥ 0

Learn how to solve the rational inequality (x+2)/(x-3) ≥ 0 step by step. This problem covers key algebraic concepts like finding critical points, creating intervals, and testing signs. Perfect for high school students (Grades 9-12).

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