https://file.aiquickdraw.com/mathbot/file/1728793456274f08vbesj.jpg

Let's analyze the two problems in the image.

### 1. Interval Notation from the Number Line:
The first part shows a number line with an open circle at \(3\) and an arrow extending to the right, covering all numbers greater than \(3\). 

This implies that the solution set does not include \(3\) itself but includes all values greater than \(3\). Hence, in interval notation:
\[
(3, \infty)
\]

### 2. Rational Inequality:
The inequality given is:
\[
\frac{1}{x+3} > \frac{1}{x-4}
\]
To solve this inequality, follow these steps:

1. **Rewrite the inequality:**
   Subtract \(\frac{1}{x-4}\) from both sides:
   \[
   \frac{1}{x+3} - \frac{1}{x-4} > 0
   \]

2. **Find a common denominator**:
   The common denominator is \((x+3)(x-4)\). So,
   \[
   \frac{(x-4) - (x+3)}{(x+3)(x-4)} > 0
   \]
   Simplifying the numerator:
   \[
   \frac{x-4 - x - 3}{(x+3)(x-4)} = \frac{-7}{(x+3)(x-4)} > 0
   \]

3. **Analyze the sign**:
   The inequality \(\frac{-7}{(x+3)(x-4)} > 0\) holds when the denominator \((x+3)(x-4)\) is negative (because \(-7\) is negative).

   Therefore, we need to determine when \((x+3)(x-4) < 0\). The key points are \(x = -3\) and \(x = 4\). So, examine the sign of \((x+3)(x-4)\) in the intervals:
   - \(x < -3\)
   - \(-3 < x < 4\)
   - \(x > 4\)

   After checking the signs, the solution occurs when \(x \in (-3, 4)\).

Thus, the solution in interval notation is:
\[
(-3, 4)
\]
Now, selecting the correct graph, graph **B** matches the interval \( (-3, 4) \), so the correct answer is **B**.

Would you like more detailed steps for solving rational inequalities?

### Relative Questions:
1. How do we graph inequalities on a number line?
2. How can we identify when to use open or closed intervals in interval notation?
3. What are the steps for solving a rational inequality?
4. How do we find the common denominator in rational inequalities?
5. What is the importance of identifying critical points in inequality problems?

### Tip:
When solving inequalities, always check the sign changes in the intervals around critical points to determine where the inequality holds.

Give the solution set, in interval notation, for the rational inequality \( \frac{1}{x+3} > \frac{1}{x-4} \) and graph the solution.

Learn how to solve the rational inequality \frac{1}{x+3} > \frac{1}{x-4} with step-by-step explanations. This problem covers key algebraic concepts such as finding a common denominator and analyzing the sign of the rational expression. Suitable for high school students (Grades 9-12).

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