Examples of Solving Rational Inequalities Problems step by step with a.determine the single rational, b.critical values, c.Number line, d.Fishball method, e.solution set, interval notation;Set notation and Graph.

### Example Problem: Solve the rational inequality 

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

We'll solve this problem step by step following the structure you've outlined.

#### Step 1: Determine the Single Rational Expression
The inequality is already in the form of a single rational expression:

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

#### Step 2: Critical Values
Critical values are the values of \( x \) that make the numerator or the denominator equal to zero because these points can either cause the rational expression to change signs or become undefined.

- **Numerator ( \( x + 2 \) ):** Set \( x + 2 = 0 \), solving gives \( x = -2 \).
- **Denominator ( \( x - 3 \) ):** Set \( x - 3 = 0 \), solving gives \( x = 3 \).

Thus, the critical values are \( x = -2 \) and \( x = 3 \).

#### Step 3: Number Line
Plot the critical values on the number line. These values divide the number line into three intervals:

- Interval 1: \( (-\infty, -2) \)
- Interval 2: \( (-2, 3) \)
- Interval 3: \( (3, \infty) \)

#### Step 4: Fishball Method (Test the Signs)
To determine the sign of the expression in each interval, pick a test point from each interval and substitute it back into the rational expression.

- **Interval 1: \( (-\infty, -2) \)**  
  Choose \( x = -3 \):  
  \[
  \frac{-3 + 2}{-3 - 3} = \frac{-1}{-6} > 0
  \]
  (The expression is positive in this interval.)

- **Interval 2: \( (-2, 3) \)**  
  Choose \( x = 0 \):  
  \[
  \frac{0 + 2}{0 - 3} = \frac{2}{-3} < 0
  \]
  (The expression is negative in this interval.)

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

#### Step 5: Solution Set
We are looking for where the expression is greater than 0, which occurs in Interval 1 and Interval 3.

- The expression is positive in \( (-\infty, -2) \) and \( (3, \infty) \).
- At \( x = -2 \), the expression equals 0 (not included in the solution set).
- At \( x = 3 \), the expression is undefined (also not included in the solution set).

Thus, the solution set is:

- **Interval Notation:** \( (-\infty, -2) \cup (3, \infty) \)
- **Set Notation:** \( \{ x \in \mathbb{R} \mid x < -2 \text{ or } x > 3 \} \)

#### Step 6: Graph
On the number line:
- Draw an open circle at \( x = -2 \) and \( x = 3 \) to indicate these points are not included.
- Shade the region to the left of \( -2 \) and the region to the right of \( 3 \).

The final graph looks like this:

```
<-----(-∞)-----(o)-----(x = -2)    (x = 3)-----(o)-----(∞)------>
```

### Summary:
1. **Single Rational Expression:** \(\frac{x + 2}{x - 3}\)
2. **Critical Values:** \( x = -2 \), \( x = 3 \)
3. **Intervals:** \( (-\infty, -2) \), \( (-2, 3) \), \( (3, \infty) \)
4. **Solution Set:**
   - Interval Notation: \( (-\infty, -2) \cup (3, \infty) \)
   - Set Notation: \( \{ x \in \mathbb{R} \mid x < -2 \text{ or } x > 3 \} \)
5. **Graph:** Open circles at \( x = -2 \) and \( x = 3 \), shaded regions on \( (-\infty, -2) \) and \( (3, \infty) \).

---

Would you like to dive deeper into any of these steps, or have any specific questions? 

Here are some additional related questions:
1. How do you solve rational inequalities with multiple factors?
2. What changes if the inequality sign is "<" instead of ">"?
3. How does the inequality change if we have more than one rational expression?
4. Can you solve rational inequalities without using a number line?
5. What is the difference between strict inequalities and non-strict inequalities in rational expressions?

**Tip:** Always remember to exclude points where the denominator is zero from your solution set, as the expression is undefined there.

Examples of Solving Rational Inequalities Problems step by step with a.determine the single rational, b.critical values, c.Number line, d.Fishball method, e.solution set, interval notation; Set notation and Graph.

Learn how to solve rational inequalities with a step-by-step guide covering critical values, number line analysis, the Fishball method, solution sets, interval notation, set notation, and graphing. Ideal for students in Grades 9-12.

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