Let's solve each inequality and provide the solutions in interval notation.

i. $|x| \geq 5$

For $|x| \geq 5$, we split it into two cases:

1. $x \geq 5$
2. $x \leq -5$

Solution: $(-\infty, -5] \cup [5, \infty)$

ii. $|x| > 1$

For $|x| > 1$, we split it into two cases:

1. $x > 1$
2. $x < -1$

Solution: $(-\infty, -1) \cup (1, \infty)$

iii. $|x + 2| > 8$

For $|x + 2| > 8$, we split it into two cases:

1. $x + 2 > 8$ → $x > 6$
2. $x + 2 < -8$ → $x < -10$

Solution: $(-\infty, -10) \cup (6, \infty)$

iv. $|x - 7| \geq 11$

For $|x - 7| \geq 11$, we split it into two cases:

1. $x - 7 \geq 11$ → $x \geq 18$
2. $x - 7 \leq -11$ → $x \leq -4$

Solution: $(-\infty, -4] \cup [18, \infty)$

v. $|x + 5| \geq 0$

Since the absolute value is always non-negative, $|x + 5| \geq 0$ holds for all $x$.

Solution: $(-\infty, \infty)$

vi. $|x - 12| > -4$

Since the absolute value is always non-negative and greater than $-4$ for all $x$, this inequality holds for all $x$.

Solution: $(-\infty, \infty)$

vii. $|2x - 5| \geq 9$

For $|2x - 5| \geq 9$, we split it into two cases:

1. $2x - 5 \geq 9$ → $2x \geq 14$ → $x \geq 7$
2. $2x - 5 \leq -9$ → $2x \leq -4$ → $x \leq -2$

Solution: $(-\infty, -2] \cup [7, \infty)$

viii. $|3x - 6| \leq 18$

For $|3x - 6| \leq 18$, we split it into two cases:

1. $3x - 6 \leq 18$ → $3x \leq 24$ → $x \leq 8$
2. $3x - 6 \geq -18$ → $3x \geq -12$ → $x \geq -4$

Solution: $[-4, 8]$

Let me know if you'd like a detailed breakdown of any specific solution or have any questions!

Related Questions

1. How would the solution change if the inequalities were strict inequalities?
2. What is the difference between solving $|x| \geq a$ and $|x| \leq a$ inequalities?
3. How do absolute value inequalities translate geometrically on a number line?
4. Can absolute value inequalities have no solution?
5. What is the interval notation for cases where there are no solutions?

Tip: When solving absolute value inequalities, always consider both the positive and negative cases to ensure all solutions are captured.