Let's solve the inequalities shown in the image.

1. Inequality: $9|x| - 9 \leq 9$

• First, isolate the absolute value term: $9|x| - 9 \leq 9$ Add 9 to both sides: $9|x| \leq 18$ Divide by 9: $|x| \leq 2$ This implies: $-2 \leq x \leq 2$ The solution is the interval: $[-2, 2]$

2. Inequality: $5|x| - 3 \geq 2$

• Isolate the absolute value term: $5|x| - 3 \geq 2$ Add 3 to both sides: $5|x| \geq 5$ Divide by 5: $|x| \geq 1$ This implies two cases: $x \geq 1 \quad \text{or} \quad x \leq -1$ So, the solution is: $(-\infty, -1] \cup [1, \infty)$

Interval Notation Summary:

1. For $9|x| - 9 \leq 9$, the interval notation is $[-2, 2]$.
2. For $5|x| - 3 \geq 2$, the interval notation is $(-\infty, -1] \cup [1, \infty]$.

Let me know if you have any questions or need further clarification!

Here are 5 related questions for further exploration:

1. How do you solve compound inequalities involving absolute values?
2. What is the difference between strict inequalities ($<$, $>$) and non-strict inequalities ($\leq$, $\geq$)?
3. How do we interpret absolute value inequalities geometrically?
4. Can absolute value inequalities have no solution?
5. How do you solve absolute value equations?

Tip: When solving absolute value inequalities, always remember to split the inequality into two cases based on the definition of absolute value.