https://file.aiquickdraw.com/mathbot/file/172901748876036b3n59n.jpg

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!

Justify each step with a property of inequalities, recall that each property listed only uses the '<' symbol but they apply for all of the inequality symbols. Solve the inequality -2x + 1 ≥ 3x + 5.

This page explains how to solve the linear inequality -2x + 1 ≥ 3x + 5 step by step using the properties of inequalities, including subtraction and division by a negative number. Suitable for students in Grades 8-10.

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