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Let's go through the problems step-by-step:

### Problem 1:

#### a. Define a variable for the unknown quantity.
- The unknown quantity here is the number of boxes of books. Let's define this as:
  - Let \( x \) be the number of boxes of books.

#### b. Write an inequality that satisfies the situation.
- The maximum total weight the elevator can carry is 1,200 pounds.
- The maintenance person weighs 180 pounds.
- Each box of books weighs 54 pounds.
- The total weight of the books plus the person must be less than or equal to 1,200 pounds. 

The inequality becomes:
\[
54x + 180 \leq 1200
\]
Where \( 54x \) represents the total weight of the boxes and \( 180 \) is the weight of the person.

#### c. Solve the inequality by showing the solve process.
1. Subtract 180 from both sides:
   \[
   54x \leq 1200 - 180
   \]
   \[
   54x \leq 1020
   \]
2. Divide both sides by 54:
   \[
   x \leq \frac{1020}{54}
   \]
   \[
   x \leq 18.89
   \]
Since the number of boxes must be an integer, \( x \leq 18 \).

#### d. State your answer in a complete sentence.
- The maintenance person can carry a maximum of 18 boxes in one trip using the elevator.

---

### Problem 2: Simplify the following expressions.

#### 2. \( 3(x - 4) + 2x \)
1. Apply the distributive property:
   \[
   3x - 12 + 2x
   \]
2. Combine like terms:
   \[
   (3x + 2x) - 12 = 5x - 12
   \]
So the expression simplifies to:
\[
5x - 12
\]
**Process used**: (C) Distribute and Combine like terms.

#### 3. \( -7x^2 + 2y + x^2 - 4x \)
1. Combine like terms:
   \[
   (-7x^2 + x^2) + 2y - 4x
   \]
   \[
   -6x^2 + 2y - 4x
   \]
So the expression simplifies to:
\[
-6x^2 + 2y - 4x
\]
**Process used**: (B) Combining like terms only.

#### 4. \( -11y^2 + 2y - 7 \)
This expression cannot be simplified further because there are no like terms.
**Process used**: (D) No simplifying possible.

#### 5. \( 4w - 2(8 - w) \)
1. Distribute the \(-2\):
   \[
   4w - 2 \cdot 8 + 2w
   \]
   \[
   4w - 16 + 2w
   \]
2. Combine like terms:
   \[
   (4w + 2w) - 16 = 6w - 16
   \]
So the expression simplifies to:
\[
6w - 16
\]
**Process used**: (C) Distribute and Combine like terms.

#### 6. \( -a + 8a - 2.5b - b \)
1. Combine like terms:
   \[
   (-a + 8a) + (-2.5b - b)
   \]
   \[
   7a - 3.5b
   \]
So the expression simplifies to:
\[
7a - 3.5b
\]
**Process used**: (B) Combining like terms only.

#### 7. \( 5(y - 3) + 2y - 10y \)
1. Distribute the \( 5 \):
   \[
   5y - 15 + 2y - 10y
   \]
2. Combine like terms:
   \[
   (5y + 2y - 10y) - 15
   \]
   \[
   -3y - 15
   \]
So the expression simplifies to:
\[
-3y - 15
\]
**Process used**: (C) Distribute and Combine like terms.

---

Let me know if you want more details or have any questions!

### Related Questions:
1. What would the inequality look like if the elevator's weight limit was 1,500 pounds?
2. How would you set up the inequality if each box weighed 60 pounds?
3. What would the expression for Problem 5 look like if the numbers were all doubled?
4. Can you create a real-life scenario for Problem 2?
5. How does the distributive property help simplify expressions like those in Problem 5?

### Tip:
When solving inequalities, remember to reverse the inequality sign if you multiply or divide by a negative number!

The sign on the elevator in a seven-story building states that the weight cannot exceed 1,200 pounds. The maintenance person's job is to move a shipment of books from the bottom floor to the 5th floor. Each box of books weighs 54 lbs. How many boxes can the person carry?

Learn how to solve an inequality problem involving an elevator weight limit and a maintenance person moving boxes of books. This problem covers key algebraic concepts, including solving inequalities and using the distributive property. Perfect for students in Grades 6-8.

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