Let's solve each of the problems step-by-step from the image.

Problem 9:

$2 \left( \frac{x}{2} - 2 \right) - 3(x + 1) =$

Step 1: Distribute the 2 and the -3: $2 \cdot \left(\frac{x}{2}\right) - 2 \cdot 2 - 3 \cdot x - 3 \cdot 1$ $= x - 4 - 3x - 3$

Step 2: Combine like terms: $(x - 3x) + (-4 - 3) = -2x - 7$

The answer to Problem 9 is: $-2x - 7$

Problem 10:

$6 \left( \frac{x}{3} + 1 \right) - 4(2x - 3) =$

Step 1: Distribute the 6 and the -4: $6 \cdot \frac{x}{3} + 6 \cdot 1 - 4 \cdot 2x + 4 \cdot 3$ $= 2x + 6 - 8x + 12$

Step 2: Combine like terms: $(2x - 8x) + (6 + 12) = -6x + 18$

The answer to Problem 10 is: $-6x + 18$

Problem 11:

$-(3x - 4) - 10\left( \frac{x}{5} + 1 \right) =$

Step 1: Distribute the -1 and the -10: $-3x + 4 - 10 \cdot \frac{x}{5} - 10 \cdot 1$ $= -3x + 4 - 2x - 10$

Step 2: Combine like terms: $(-3x - 2x) + (4 - 10) = -5x - 6$

The answer to Problem 11 is: $-5x - 6$

Problem 12:

$\frac{1}{2}(a + 2b) - \frac{1}{3}(3a - b) =$

Step 1: Distribute the $\frac{1}{2}$ and $\frac{1}{3}$: $\frac{1}{2}a + \frac{1}{2}(2b) - \frac{1}{3}(3a) + \frac{1}{3}b$ $= \frac{1}{2}a + b - a + \frac{1}{3}b$

Step 2: Combine like terms: $\left(\frac{1}{2}a - a\right) + \left(b + \frac{1}{3}b\right) = -\frac{1}{2}a + \frac{4}{3}b$

The answer to Problem 12 is: $-\frac{1}{2}a + \frac{4}{3}b$

Problem 13:

$-\frac{1}{4}(5x - 3y) + y =$

Step 1: Distribute the $-\frac{1}{4}$: $-\frac{1}{4} \cdot 5x + \frac{1}{4} \cdot 3y + y$ $= -\frac{5}{4}x + \frac{3}{4}y + y$

Step 2: Combine like terms: $-\frac{5}{4}x + \left( \frac{3}{4}y + y \right) = -\frac{5}{4}x + \frac{7}{4}y$

The answer to Problem 13 is: $-\frac{5}{4}x + \frac{7}{4}y$

Problem 14:

$3x - 6\left( \frac{x}{2} + \frac{1}{3} \right) - 4(x - 1) =$

Step 1: Distribute the -6 and the -4: $3x - 6 \cdot \frac{x}{2} - 6 \cdot \frac{1}{3} - 4x + 4$ $= 3x - 3x - 2 - 4x + 4$

Step 2: Combine like terms: $(3x - 3x - 4x) + (-2 + 4) = -4x + 2$

The answer to Problem 14 is: $-4x + 2$

Let me know if you'd like more details on any of these, or if you have any questions!

Here are 5 related questions for you to explore:

1. How would you solve similar problems if fractions were more complex?
2. Can you generalize the steps for distributing negative signs in algebraic expressions?
3. How do coefficients affect the simplification of variables in equations?
4. What are the common pitfalls when combining like terms with fractions?
5. How can you check your final answer for accuracy in algebraic simplifications?

Tip: Always distribute carefully, especially with negative signs, to avoid common errors in simplification.