https://file.aiquickdraw.com/mathbot/file/1729903048835s8axy3e4.jpg

To solve the inequality \( \frac{x - 4}{x - 7} > 0 \), we can use the graph provided to determine when the expression \( \frac{x - 4}{x - 7} \) is positive. 

### Step-by-Step Solution

1. **Identify the critical points**:
   - The expression \( \frac{x - 4}{x - 7} \) becomes zero when \( x = 4 \) (numerator is zero).
   - The expression is undefined at \( x = 7 \) (denominator is zero).

   These points, \( x = 4 \) and \( x = 7 \), are crucial because they separate intervals where the expression may change signs.

2. **Analyze the intervals**:
   Based on the critical points, we have three intervals to consider:
   - \( (-\infty, 4) \)
   - \( (4, 7) \)
   - \( (7, \infty) \)

3. **Determine the sign of \( \frac{x - 4}{x - 7} \) in each interval**:
   - For \( x \in (-\infty, 4) \): Choose a test point, like \( x = 0 \). Substituting into \( \frac{x - 4}{x - 7} \), we get \( \frac{0 - 4}{0 - 7} = \frac{-4}{-7} = \frac{4}{7} \), which is positive.
   - For \( x \in (4, 7) \): Choose a test point, like \( x = 5 \). Substituting into \( \frac{x - 4}{x - 7} \), we get \( \frac{5 - 4}{5 - 7} = \frac{1}{-2} = -\frac{1}{2} \), which is negative.
   - For \( x \in (7, \infty) \): Choose a test point, like \( x = 8 \). Substituting into \( \frac{x - 4}{x - 7} \), we get \( \frac{8 - 4}{8 - 7} = \frac{4}{1} = 4 \), which is positive.

4. **Conclusion**:
   The inequality \( \frac{x - 4}{x - 7} > 0 \) holds for intervals where the expression is positive, which are:
   \[
   (-\infty, 4) \cup (7, \infty)
   \]

### Solution Set
Thus, the solution set in interval notation is:
\[
(-\infty, 4) \cup (7, \infty)
\]

Would you like further clarification on any step?

---

#### Related Questions
1. How would the solution change if the inequality were \( \frac{x - 4}{x - 7} < 0 \)?
2. What happens if \( x = 4 \) or \( x = 7 \) were included in the inequality?
3. How can we verify the solution using another method, like a sign chart?
4. How does the graph reflect the behavior of the function near \( x = 4 \) and \( x = 7 \)?
5. What are asymptotes, and why does \( x = 7 \) create a vertical asymptote?

#### Tip
Always test intervals around critical points when solving rational inequalities to check for changes in sign.

State the solution set to the inequality \( \frac{x - 4}{x - 7} > 0 \) by reading the graph provided.

Learn how to find the solution set to the inequality \( \frac{x - 4}{x - 7} > 0 \) by analyzing a graph. This solution involves identifying critical points, testing intervals, and understanding the behavior of rational functions. Suitable for high school students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation