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 $4x + 30$.

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.