The problem you're working on involves solving the absolute value inequality:

$|3x - 2| > 8$

To solve this, we rewrite the absolute value inequality as two separate linear inequalities:

1. $3x - 2 > 8$
2. $3x - 2 < -8$

Now let's solve both inequalities separately:

1. $3x - 2 > 8$

Add 2 to both sides: $3x > 10$ Now divide by 3: $x > \frac{10}{3}$

2. $3x - 2 < -8$

Add 2 to both sides: $3x < -6$ Now divide by 3: $x < -2$

Thus, the two linear inequalities are:

• $x > \frac{10}{3}$
• $x < -2$

Since the original inequality is "greater than" ( $|3x - 2| > 8$ ), the solution represents values that are outside of the interval between $-2$ and $\frac{10}{3}$. Therefore, the two inequalities should be connected by "or," meaning:

$x < -2 \quad \text{or} \quad x > \frac{10}{3}$

Summary:

• Inequalities: $x < -2$ and $x > \frac{10}{3}$
• Connect with "or" (since we're looking for values outside the interval).

Would you like more details or explanations on any of these steps?

Here are five related questions to deepen your understanding:

1. How do you solve absolute value inequalities that involve "less than" ($|expression| < value$)?
2. Can you graph the solution of $|3x - 2| > 8$ on a number line?
3. What is the difference between "and" and "or" in compound inequalities?
4. How would the solution change if the inequality were $|3x - 2| \geq 8$ instead of strictly greater than?
5. How does the behavior of absolute value inequalities change when negative values are involved?

Tip: Always remember that "or" is used for inequalities involving "greater than" cases with absolute values, and "and" is used for "less than" cases.