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Let's start by analyzing the problems shown in the image you uploaded. The first part (Problem 16) asks to give a property of real numbers (associative, commutative, or distributive) or a property of equality (addition or multiplication) that justifies each step in solving the equation.

### Problem 16 Solution Explanation:
The equation provided is:
\[
5(x+3) + 2x = 4x + 9
\]

Here are the steps with the corresponding properties:

1. **Step 1: \(5(x + 3) + 2x = 4x + 9\) → \(5x + 15 + 2x = 4x + 9\)**
   - **Property:** Distributive Property (distributing \(5\) over \(x + 3\)).

2. **Step 2: \(5x + 15 + 2x = 4x + 9\) → \(7x + 15 = 4x + 9\)**
   - **Property:** Commutative Property of Addition (adding \(5x\) and \(2x\)).

3. **Step 3: \(7x + 15 = 4x + 9\)**
   - **Property:** Associative Property of Addition (grouping terms).

4. **Step 4: \(7x + 15 - 4x = 4x + 9 - 4x\) → \(7x - 4x + 15 = 9\)**
   - **Property:** Subtraction Property of Equality (subtracting \(4x\) from both sides).

5. **Step 5: \(3x + 15 = 9\)**
   - **Property:** Simplification (simplifying \(7x - 4x\)).

6. **Step 6: \(3x + 15 - 15 = 9 - 15\) → \(3x = -6\)**
   - **Property:** Subtraction Property of Equality (subtracting \(15\) from both sides).

7. **Step 7: \(\frac{1}{3} \cdot 3x = \frac{1}{3} \cdot -6\) → \(x = -2\)**
   - **Property:** Multiplication Property of Equality (multiplying by \(\frac{1}{3}\) to solve for \(x\)).

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### Problem 17: Integer Problem

#### Part (a)
You are given the problem about two consecutive even integers:
"When the smaller integer is added to three times the larger integer, the result is two less than five times the smaller integer."

Rafael's attempt:
\[ n + 3n + 2 = 5n - 2 \]

Rafael made an error because the larger integer should be written as \(n + 2\) instead of \(n\) or \(n + 1\). The correct equation should represent two *consecutive even* integers, meaning if the smaller integer is \(n\), the larger one is \(n + 2\).

#### Part (b)
We correct the equation:
\[
n + 3(n + 2) = 5n - 2
\]
Now, solve this:
\[
n + 3n + 6 = 5n - 2
\]
Simplify:
\[
4n + 6 = 5n - 2
\]
Subtract \(4n\) from both sides:
\[
6 = n - 2
\]
Add \(2\) to both sides:
\[
n = 8
\]
Thus, the smaller integer is \(n = 8\), and the larger integer is \(n + 2 = 10\).

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Would you like further details or explanations on any step? 

Here are five related questions you could explore:
1. Can you explain the difference between associative and commutative properties?
2. What are some real-world applications of these properties of equality?
3. How do you handle equations involving consecutive odd integers instead of even integers?
4. Could you derive the solution using a different strategy, such as using substitution?
5. How would you explain the difference between multiplication and addition properties of equality to a peer?

**Tip:** When working with consecutive integers, carefully define what type of integers you're dealing with (even, odd, etc.) to avoid mistakes.

Give a property of real numbers (associative, commutative, or distributive) or a property of equality (addition or multiplication) that justifies each step in the solution of the equation.

Learn how to solve algebraic equations by applying the distributive, commutative, and associative properties, along with the properties of equality. This problem guides students through justifying each step using algebraic concepts. Suitable for students in Grades 8-10.

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