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In this image, you provided a list of inequality problems and a worked solution for the first one. The problem is to determine the solution set of inequalities. Let me break down the provided solution for the first inequality and give an overview of how the other inequalities might be solved.

### Solving the first inequality:

1. **Given inequality**:
   \[
   x^2 - 5x + 6 > 0
   \]

2. **Factoring the quadratic expression**:
   \[
   (x - 2)(x - 3) > 0
   \]
   This shows that the roots are \( x = 2 \) and \( x = 3 \), so these are the points where the inequality changes sign.

3. **Determining intervals**:
   The roots divide the number line into three intervals:
   - \( (-\infty, 2) \)
   - \( (2, 3) \)
   - \( (3, \infty) \)

4. **Sign analysis**:
   By checking the sign of the expression \( (x - 2)(x - 3) \) in each interval:
   - For \( x < 2 \), the expression is positive.
   - For \( 2 < x < 3 \), the expression is negative.
   - For \( x > 3 \), the expression is positive.

5. **Solution**:
   The inequality \( (x - 2)(x - 3) > 0 \) holds when \( x < 2 \) or \( x > 3 \). Therefore, the solution set is:
   \[
   (-\infty, 2) \cup (3, \infty)
   \]

This solution is confirmed by the graph in the image, which shows the intervals where the inequality is valid.

---

### General Strategy for the Other Inequalities:

1. **Rational Inequalities** (e.g., #2, #5, #7, #9, #10):
   - Solve the corresponding equation by finding the critical points where the numerator or denominator is zero.
   - Determine the sign of the expression in the intervals between these points.

2. **Compound Inequalities** (e.g., #3, #9):
   - Break the inequality into two parts, solve each separately, and then find the intersection of the solution sets.

3. **Inequalities with Fractions** (e.g., #6, #8):
   - Multiply through by the denominators, ensuring to reverse the inequality sign if multiplying by a negative.

4. **Absolute Value Inequalities** (e.g., #4):
   - Break it into two cases depending on whether the expression inside the absolute value is positive or negative.

Would you like to solve the next inequality in detail, or do you have any questions about this process?

---

### Related Questions:
1. How do you solve rational inequalities that involve multiple factors in the denominator?
2. What is the method for solving compound inequalities like the third one?
3. How does the solution method change when absolute values are involved in an inequality?
4. What are the steps to solve quadratic inequalities using the sign test on intervals?
5. How do you graphically represent the solution sets of inequalities?

### Tip:
Always check the points where the expression becomes zero or undefined, as these divide the number line into regions that determine the sign of the inequality.

Determine the solution set of the inequality x^2 - 5x + 6 > 0

Learn how to solve the quadratic inequality x^2 - 5x + 6 > 0 using factoring and sign analysis. This problem covers key algebraic concepts such as quadratic expressions and solving inequalities. Suitable for students in Grades 9-11.

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