We are given the absolute value inequality:

$5 |x - 3| + 8 \geq 18$

Step 1: Isolate the Absolute Value

Subtract 8 from both sides:

$5 |x - 3| \geq 10$

Now, divide by 5:

$|x - 3| \geq 2$

Step 2: Solve for $x$

The absolute value inequality $|x - 3| \geq 2$ means that the expression inside the absolute value is either:

1. $x - 3 \geq 2$, or
2. $x - 3 \leq -2$

Solving each:

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

Step 3: Write the Final Solution

The solution set is:

$x \leq 1 \quad \text{or} \quad x \geq 5$

In interval notation:

$(-\infty, 1] \cup [5, \infty)$

Would you like a graphical representation or further explanations?

Related Questions

1. How do you solve absolute value inequalities with less than ($\leq$) instead of greater than ($\geq$)?
2. What is the difference between "or" and "and" in absolute value inequalities?
3. How do you graph the solution set on a number line?
4. Can this inequality be solved using another method?
5. How would the solution change if it were $5|x - 3| + 8 < 18$?

Tip

When solving absolute value inequalities, remember:

• $|A| < B$ leads to a conjunction ("and"), meaning $-B < A < B$.
• $|A| > B$ leads to a disjunction ("or"), meaning $A < -B$ or $A > B$.