To solve the given absolute value inequality:

$2|x - 5| \geq 8$

Step-by-Step Solution:

1. Isolate the absolute value term:
   Divide both sides of the inequality by 2: $|x - 5| \geq 4$

2. Interpret the absolute value inequality:
   The absolute value inequality $|A| \geq B$ (where $B > 0$) means: $A \geq B \quad \text{or} \quad A \leq -B.$ Here $A = x - 5$ and $B = 4$.
   Thus: $x - 5 \geq 4 \quad \text{or} \quad x - 5 \leq -4.$

3. Solve each inequality:

   • For $x - 5 \geq 4$:
     Add 5 to both sides: $x \geq 9.$
   • For $x - 5 \leq -4$:
     Add 5 to both sides: $x \leq 1.$

4. Combine the solutions:
   The solution is: $x \leq 1 \quad \text{or} \quad x \geq 9.$

Final Answer:

$x \in (-\infty, 1] \cup [9, \infty)$

Graphical Interpretation:

This solution means that $x$ is either less than or equal to 1, or greater than or equal to 9.

Would you like me to show this graphically or explain further? Let me know!

Related Questions:

1. How would the solution change if the inequality were $2|x - 5| < 8$?
2. What if the inequality had $\leq$ instead of $\geq$?
3. How do you solve $|x + 2| \geq 6$?
4. What are the steps to solve $|2x - 3| > 5$?
5. How can you write $|x - a| \geq b$ in interval notation?

Tip:

Always isolate the absolute value term first before splitting the inequality into two cases.