I see the uploaded worksheet. Let me transcribe the problems and solve them using the distributive property. I'll provide detailed solutions for each question.

Problems

1. $8x(5) - 2(x + 1)$
2. $-\frac{1}{4} + \frac{2}{3} + \frac{1}{3} - 3$
3. $3(x + 2y - 4x)$
4. $-\frac{1}{2}(4x - 8) + 5x$
5. $2.4x + 3.2(3 + x)$
6. $-\frac{1}{3}(x + y) + \frac{2}{3}(x + y)$
7. $\frac{3}{4}(4x - 8) + \frac{2}{5}(10x - 15)$

Solutions

1. $8x(5) - 2(x + 1)$

Apply the distributive property: $8x \cdot 5 - 2(x) - 2(1) = 40x - 2x - 2$ Combine like terms: $38x - 2$

2. $-\frac{1}{4} + \frac{2}{3} + \frac{1}{3} - 3$

Simplify fractions: $-\frac{1}{4} + \frac{3}{3} - 3 = -\frac{1}{4} + 1 - 3 = -\frac{1}{4} - 2$ Convert $-2$ to a fraction with denominator 4: $-\frac{1}{4} - \frac{8}{4} = -\frac{9}{4}$

3. $3(x + 2y - 4x)$

Distribute $3$: $3 \cdot x + 3 \cdot 2y - 3 \cdot 4x = 3x + 6y - 12x$ Combine like terms: $-9x + 6y$

4. $-\frac{1}{2}(4x - 8) + 5x$

Distribute $-\frac{1}{2}$: $-\frac{1}{2} \cdot 4x + -\frac{1}{2} \cdot -8 + 5x = -2x + 4 + 5x$ Combine like terms: $3x + 4$

5. $2.4x + 3.2(3 + x)$

Distribute $3.2$: $2.4x + 3.2 \cdot 3 + 3.2 \cdot x = 2.4x + 9.6 + 3.2x$ Combine like terms: $5.6x + 9.6$

6. $-\frac{1}{3}(x + y) + \frac{2}{3}(x + y)$

Distribute the fractions: $-\frac{1}{3}x - \frac{1}{3}y + \frac{2}{3}x + \frac{2}{3}y$ Combine like terms: $\left(-\frac{1}{3}x + \frac{2}{3}x\right) + \left(-\frac{1}{3}y + \frac{2}{3}y\right) = \frac{1}{3}x + \frac{1}{3}y$

7. $\frac{3}{4}(4x - 8) + \frac{2}{5}(10x - 15)$

Distribute the fractions: $\frac{3}{4} \cdot 4x - \frac{3}{4} \cdot 8 + \frac{2}{5} \cdot 10x - \frac{2}{5} \cdot 15$ Simplify: $3x - 6 + 4x - 6$ Combine like terms: $7x - 12$

Let me know if you'd like detailed explanations of any step or further clarifications!

Related Questions:

1. What are some additional examples of the distributive property?
2. How can you identify when to use the distributive property?
3. What are some common mistakes when simplifying expressions like these?
4. How does combining like terms affect the result of these problems?
5. Why is the distributive property useful in algebra?

Tip: Always distribute multiplication across addition or subtraction before combining like terms!