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Let's go through the problems one by one, and I'll help solve them using the distributive property.

### Problem 1: \( 5(x - 7) \)
Using the distributive property:
\[
5(x - 7) = 5 \cdot x - 5 \cdot 7 = 5x - 35
\]
So, the solution is \( 5x - 35 \).

### Problem 2: \( 7(-8x + 5) \)
\[
7(-8x + 5) = 7 \cdot (-8x) + 7 \cdot 5 = -56x + 35
\]
The answer is \( -56x + 35 \).

### Problem 3: \( -1(4x - 26) \)
\[
-1(4x - 26) = -1 \cdot 4x + (-1) \cdot (-26) = -4x + 26
\]
So, the answer is \( -4x + 26 \).

### Problem 4: \( \frac{1}{3}(24x - 54) \)
Distribute \( \frac{1}{3} \):
\[
\frac{1}{3}(24x - 54) = \frac{1}{3} \cdot 24x - \frac{1}{3} \cdot 54 = 8x - 18
\]
The answer is \( 8x - 18 \).

### Problem 5: \( 4(3x - 2y + 9) \)
Using the distributive property:
\[
4(3x - 2y + 9) = 4 \cdot 3x + 4 \cdot (-2y) + 4 \cdot 9 = 12x - 8y + 36
\]
So, the answer is \( 12x - 8y + 36 \).

### Problem 6: \( 23 + 3(6x - 7) \)
First, apply the distributive property to \( 3(6x - 7) \):
\[
3(6x - 7) = 18x - 21
\]
Now add 23:
\[
23 + 18x - 21 = 18x + 2
\]
The answer is \( 18x + 2 \).

### Problem 7: Double the expression \( 2(Hx - 5.75) \)
To double the expression, multiply everything by 2:
\[
2(Hx - 5.75) = 2 \cdot Hx - 2 \cdot 5.75 = 2Hx - 11.5
\]
So, the answer is \( 2Hx - 11.5 \).

### Problem 8: \( -2 + 2 \left( \frac{1}{2}x - \frac{1}{4} \right) \)
First, apply the distributive property:
\[
2 \left( \frac{1}{2}x - \frac{1}{4} \right) = 2 \cdot \frac{1}{2}x - 2 \cdot \frac{1}{4} = x - \frac{1}{2}
\]
Now subtract 2:
\[
-2 + x - \frac{1}{2} = x - 2.5
\]
The answer is \( x - 2.5 \).

### Problem 9: \( -14(9 + 0.7x) \)
Using the distributive property:
\[
-14(9 + 0.7x) = -14 \cdot 9 + (-14) \cdot 0.7x = -126 - 9.8x
\]
So, the answer is \( -126 - 9.8x \).

### Problem 10: \( 6 - 8(2x + 5) \)
First, apply the distributive property:
\[
-8(2x + 5) = -8 \cdot 2x - 8 \cdot 5 = -16x - 40
\]
Now subtract from 6:
\[
6 - 16x - 40 = -16x - 34
\]
The answer is \( -16x - 34 \).

### Problem 11: If the length of a square is \( x + 7 \frac{1}{2} \), what is the perimeter?
The perimeter of a square is 4 times the length of one side. The side length is \( x + 7 \frac{1}{2} \), or \( x + 7.5 \).
\[
\text{Perimeter} = 4(x + 7.5) = 4x + 30
\]
So, the perimeter is \( 4x + 30 \).

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Feel free to ask for further clarification! Here are five related questions:

1. How does the distributive property work for both addition and subtraction?
2. How can you simplify expressions involving fractions and the distributive property?
3. How does multiplying a binomial affect each term inside parentheses?
4. What is the importance of correctly distributing negative signs across terms?
5. How do you find the perimeter of a rectangle or square in terms of algebraic expressions?

**Tip:** When using the distributive property, always ensure each term inside the parentheses is multiplied by the factor outside.

Help solve distributive property problems.

Solve problems using the distributive property with detailed step-by-step solutions. This worksheet covers essential algebraic concepts like simplifying expressions and applying the distributive property, suitable for students in Grades 6-8.

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