Let's walk through the steps and match them with the correct properties of inequalities from the list provided in the image:

1. Initial inequality: $-2x + 1 \geq 3x + 5$

2. Step 1 to 2: We subtract $3x$ from both sides: $-2x - 3x + 1 \geq 5$ $-5x + 1 \geq 5$ This step uses Property A: If $A \leq B$, then $A - C \leq B - C$, because subtracting a constant from both sides maintains the inequality.

3. Step 2 to 3: We subtract 1 from both sides: $-5x + 1 - 1 \geq 5 - 1$ $-5x \geq 4$ This step also uses Property A, where subtracting the same constant from both sides preserves the inequality.

4. Step 3 to 4: Now we divide both sides by $-5$, and since we are dividing by a negative number, we reverse the inequality: $x \leq \frac{4}{-5}$ $x \leq -\frac{4}{5}$ This step uses Property C: If $A \leq B$ and $C < 0$, then $AC \geq BC$, because multiplying or dividing by a negative number reverses the inequality.

Final Steps with Justification:

1. Step 1 to 2: Property A (Subtracting $3x$ from both sides)
2. Step 2 to 3: Property A (Subtracting 1 from both sides)
3. Step 3 to 4: Property C (Dividing by a negative number reverses the inequality)

Let me know if you'd like further details or clarifications on any of these steps!

Relative Questions:

1. What happens to an inequality when you multiply or divide by a negative number?
2. How does Property A work when you subtract a variable from both sides of the inequality?
3. Can you apply Property D in any of the steps? Why or why not?
4. Why is it important to reverse the inequality when dividing by a negative number?
5. What is the difference between Properties B and C, and how do they apply to different scenarios?

Tip: Always remember to reverse the inequality when multiplying or dividing by a negative number!