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: $x \in [-1, -1] \cup [3, 4)$

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.