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 $x$).

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 $n + 2 = 10$.

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.