Let's solve both questions one at a time.

Question 1:

The equation is $\frac{1}{2} |x + 2| = 10$.

Steps:

1. Multiply both sides by 2 to eliminate the fraction: $|x + 2| = 20$
2. To solve for $x$, recall that the absolute value equation $|A| = B$ leads to two possible equations: $x + 2 = 20 \quad \text{or} \quad x + 2 = -20$
3. Solve each:
   • For $x + 2 = 20$: $x = 18$
   • For $x + 2 = -20$: $x = -22$

Thus, the solutions are $x = 18$ and $x = -22$.

Answer: The correct choices are A (x = -22) and F (x = 18).

Question 2:

The equation is $\frac{1}{3} |2x - 5| + 1 = -1$.

Steps:

1. Subtract 1 from both sides: $\frac{1}{3} |2x - 5| = -2$
2. Multiply both sides by 3 to eliminate the fraction: $|2x - 5| = -6$
3. Notice that an absolute value expression cannot be negative. Therefore, there are no solutions to this equation.

Answer: The correct choice is F (No Solutions).

Would you like more details on any of the steps? Here are 5 related questions for further practice:

1. How do you solve absolute value inequalities?
2. What are the steps to graph solutions on a number line?
3. How do you solve equations involving absolute values with variables on both sides?
4. Can absolute value functions ever produce negative outputs?
5. What happens when you square both sides of an absolute value equation?

Tip: Always check for extraneous solutions when solving absolute value equations, especially after squaring both sides or multiplying.