https://file.aiquickdraw.com/mathbot/file/1726555213065t5ppsr9b.jpg

I can help you solve the exercises in the image. I'll work through them one by one.

### Let's start with the first two problems:
1. **\( 2 \leq x^2 - x < 6 \)**

To solve this inequality:

- Split into two parts: 
  \[
  2 \leq x^2 - x \quad \text{and} \quad x^2 - x < 6
  \]

- Solve the first part:
  \[
  2 \leq x^2 - x \quad \Rightarrow \quad x^2 - x - 2 \geq 0
  \]
  Factor the quadratic:
  \[
  (x - 2)(x + 1) \geq 0
  \]
  The critical points are \(x = 2\) and \(x = -1\). Using a sign chart, we find that the solution is \(x \in (-\infty, -1] \cup [2, \infty)\).

- Solve the second part:
  \[
  x^2 - x < 6 \quad \Rightarrow \quad x^2 - x - 6 < 0
  \]
  Factor the quadratic:
  \[
  (x - 3)(x + 2) < 0
  \]
  The critical points are \(x = 3\) and \(x = -2\). Using a sign chart, the solution is \(x \in (-2, 3)\).

- Now, combine the two results:
  \[
  (-\infty, -1] \cup [2, \infty) \quad \text{and} \quad (-2, 3)
  \]
  The intersection gives the solution:
  \[
  x \in [-1, -1] \cup [2, 3)
  \]

2. **\( 3 \leq x^2 - 2x < 8 \)**

To solve this inequality:

- Split into two parts: 
  \[
  3 \leq x^2 - 2x \quad \text{and} \quad x^2 - 2x < 8
  \]

- Solve the first part:
  \[
  3 \leq x^2 - 2x \quad \Rightarrow \quad x^2 - 2x - 3 \geq 0
  \]
  Factor the quadratic:
  \[
  (x - 3)(x + 1) \geq 0
  \]
  The critical points are \(x = 3\) and \(x = -1\). Using a sign chart, we find that the solution is \(x \in (-\infty, -1] \cup [3, \infty)\).

- Solve the second part:
  \[
  x^2 - 2x < 8 \quad \Rightarrow \quad x^2 - 2x - 8 < 0
  \]
  Factor the quadratic:
  \[
  (x - 4)(x + 2) < 0
  \]
  The critical points are \(x = 4\) and \(x = -2\). Using a sign chart, the solution is \(x \in (-2, 4)\).

- Now, combine the two results:
  \[
  (-\infty, -1] \cup [3, \infty) \quad \text{and} \quad (-2, 4)
  \]
  The intersection gives the solution:
  \[
  x \in [-1, -1] \cup [3, 4)
  \]

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Would you like me to continue solving the next problems, or would you prefer further clarification on these? Here are some questions that might expand your understanding:

1. How do you determine the sign of intervals in quadratic inequalities?
2. What is a sign chart, and how is it useful in solving inequalities?
3. How do we factor quadratic expressions like \(x^2 - 2x - 3\)?
4. What is the process of combining the results of inequalities?
5. How would the solution change if we had strict inequalities (i.e., \(<\) or \(>\))?

### Tip:
When solving quadratic inequalities, always check the signs of the factors in each interval between critical points. This will help you determine where the inequality holds.

Solve the set of inequalities and express each solution in interval notation.

This exercise involves solving a set of inequalities including linear, quadratic, and rational expressions. Learn how to solve these inequalities, express the solutions in interval notation, and sketch their graphs. This is suitable for students in Grades 9-12, covering key algebraic concepts.

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