To determine which inequalities can be solved using only the Addition Property of Inequality, let’s examine each option and identify whether adding or subtracting a term will isolate the variable without needing any other operations, such as multiplication or division.

Analysis of Each Option:

1. Option A: $w - 4 > -10$

   • To isolate $w$, we can add $4$ to both sides: $w - 4 + 4 > -10 + 4 \Rightarrow w > -6$
   • This inequality can indeed be solved using only addition.

2. Option B: $12 \geq 20x$

   • To isolate $x$, we would need to divide both sides by $20$, which is not allowed using only the Addition Property of Inequality.
   • This inequality cannot be solved using only addition.

3. Option C: $\frac{7y}{10} \leq 5.5$

   • To isolate $y$, we would need to multiply both sides by $\frac{10}{7}$, not allowed by the Addition Property of Inequality.
   • This inequality cannot be solved using only addition.

4. Option D: $-3.25 < x - 9.75$

   • To isolate $x$, we can add $9.75$ to both sides: $-3.25 + 9.75 < x - 9.75 + 9.75 \Rightarrow 6.5 < x$
   • This inequality can indeed be solved using only addition.

5. Option E: $z - \frac{1}{2} \geq 9.8$

   • To isolate $z$, we can add $\frac{1}{2}$ to both sides: $z - \frac{1}{2} + \frac{1}{2} \geq 9.8 + \frac{1}{2} \Rightarrow z \geq 10.3$
   • This inequality can be solved using only addition.

Conclusion:

The inequalities that can be solved using only the Addition Property of Inequality are:

• A: $w - 4 > -10$
• D: $-3.25 < x - 9.75$
• E: $z - \frac{1}{2} \geq 9.8$

Would you like a deeper explanation for any of these steps?

Here are five additional questions that build on this topic:

1. How does the Addition Property of Inequality differ from the Multiplication Property of Inequality?
2. Can subtraction also be considered part of the Addition Property of Inequality?
3. Why can’t inequalities involving division be solved with the Addition Property alone?
4. How would the solution change if the inequality had a negative coefficient on the variable?
5. In what cases would you use the Multiplication Property of Inequality instead?

Tip: Remember, the Addition Property of Inequality states that you can add (or subtract) the same value from both sides of an inequality without changing its direction.