Let's tackle the problems one by one:

Question 5

Find the absolute value inequality which results in $x \leq -8$ or $2 \leq x$:

This inequality represents two conditions: $x \leq -8$ and $x \geq 2$. The absolute value inequality that fits this situation is one that separates the solution into two distinct intervals. The solution corresponds to the format $|x + c| \geq k$, where the result includes values less than or equal to a certain point or greater than or equal to another point.

Thus, the correct choice is:

• a. $|x + 3| + 5 \geq 10$

Explanation: Simplifying $|x + 3| + 5 \geq 10$ gives $|x + 3| \geq 5$. This inequality is satisfied when $x + 3 \leq -5$ or $x + 3 \geq 5$, which simplifies to $x \leq -8$ or $x \geq 2$, matching the given condition.

Question 6

Solve the inequality: $|4x + 6| < 10$:

Step-by-step:

1. First, express the inequality without absolute values: $-10 < 4x + 6 < 10$
2. Subtract 6 from all sides: $-16 < 4x < 4$
3. Divide by 4: $-4 < x < 1$ Thus, the solution is the interval $(-4, 1)$.

The correct choice is:

• a. $(-4, 1)$

Question 7

Solve the inequality: $|5x + 2| \geq 7$:

Step-by-step:

1. Write two inequalities: $5x + 2 \geq 7 \quad \text{or} \quad 5x + 2 \leq -7$
2. Solve each:
   • For $5x + 2 \geq 7$: $5x \geq 5 \quad \Rightarrow \quad x \geq 1$
   • For $5x + 2 \leq -7$: $5x \leq -9 \quad \Rightarrow \quad x \leq -\frac{9}{5}$

Thus, the solution is $x \geq 1$ or $x \leq -\frac{9}{5}$, which corresponds to $(-\infty, -\frac{9}{5}] \cup [1, \infty)$.

The correct choice is:

• a. $(-\infty, -\frac{9}{5}] \cup [1, \infty)$

Question 8

Solve the inequality: $|x - 5| - 7 \leq 8$:

Step-by-step:

1. Simplify the inequality: $|x - 5| \leq 15$
2. Write the inequality without the absolute value: $-15 \leq x - 5 \leq 15$
3. Add 5 to all sides: $-10 \leq x \leq 20$ Thus, the solution is the interval $[-10, 20]$.

The correct choice is:

• a. $[-10, 20]$

Question 9

Solve the inequality: $|3x + 6| - 10 < 17$:

Step-by-step:

1. Simplify the inequality: $|3x + 6| < 27$
2. Write the inequality without the absolute value: $-27 < 3x + 6 < 27$
3. Subtract 6 from all sides: $-33 < 3x < 21$
4. Divide by 3: $-11 < x < 7$ Thus, the solution is the interval $(-11, 7)$.

The correct choice is:

• a. $(-11, 7)$

Question 10

Solve the inequality: $|x| \leq 4$:

Step-by-step:

1. The absolute value inequality $|x| \leq 4$ means $-4 \leq x \leq 4$, which represents the closed interval $[-4, 4]$.

The correct choice is:

• b. $[-4, 4]$

Do you have any questions or would you like more details?

Here are 5 related questions:

1. How do you determine when an absolute value inequality represents a union of intervals?
2. Can you explain the difference between "less than" and "less than or equal to" for absolute values?
3. How do we solve absolute value inequalities that include shifts and scaling like $|2x + 1|$?
4. Why do some absolute value inequalities result in two intervals while others result in one?
5. What happens when the inequality has absolute values on both sides?

Tip: Always isolate the absolute value expression first before handling the inequality.