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Let's go through each question and solve them step by step:

### 1. Expand the brackets in each of these expressions.

#### a. \( 6(x + 4) \)
\[
6(x + 4) = 6 \cdot x + 6 \cdot 4 = 6x + 24
\]

#### b. \( \frac{1}{2}(4 - 6x) \)
\[
\frac{1}{2}(4 - 6x) = \frac{1}{2} \cdot 4 - \frac{1}{2} \cdot 6x = 2 - 3x
\]

#### c. \( \frac{2}{3}(6x + 9) \)
\[
\frac{2}{3}(6x + 9) = \frac{2}{3} \cdot 6x + \frac{2}{3} \cdot 9 = 4x + 6
\]

#### d. \( \frac{6(8x - 10)}{4} \)
\[
\frac{6(8x - 10)}{4} = \frac{6 \cdot 8x - 6 \cdot 10}{4} = \frac{48x - 60}{4} = 12x - 15
\]

---

### 2. Simplify these expressions.

#### a. \( 9(x + 2) + 3(4x - 1) \)
\[
9(x + 2) + 3(4x - 1) = 9x + 18 + 12x - 3 = (9x + 12x) + (18 - 3) = 21x + 15
\]

#### b. \( 5(3 - 2x) + 4(-7x - 4) \)
\[
5(3 - 2x) + 4(-7x - 4) = 15 - 10x - 28x - 16 = 15 - 44x - 16 = -44x - 1
\]

#### c. \( 6(2x - 5) - 7(3 - x) \)
\[
6(2x - 5) - 7(3 - x) = 12x - 30 - 21 + 7x = (12x + 7x) + (-30 - 21) = 19x - 51
\]

#### d. \( -9(3x - 1) - (10 - 4x) \)
\[
-9(3x - 1) - (10 - 4x) = -27x + 9 - 10 + 4x = (-27x + 4x) + (9 - 10) = -23x - 1
\]

---

### 3. Simplify these expressions.

#### a. \( \frac{12x - 8}{4} - 3(x - 5) \)
\[
\frac{12x - 8}{4} - 3(x - 5) = (3x - 2) - (3x - 15) = 3x - 2 - 3x + 15 = -2 + 15 = 13
\]

#### b. \( \frac{3}{2}(14x - 8) - \frac{18x + 6}{3} \)
\[
\frac{3}{2}(14x - 8) - \frac{18x + 6}{3} = \frac{3}{2} \cdot 14x - \frac{3}{2} \cdot 8 - 6x - 2 = 21x - 12 - 6x - 2 = (21x - 6x) + (-12 - 2) = 15x - 14
\]

#### c. \( \frac{12(4x - 3)}{15} + \frac{10(6x - 3)}{15} \)
\[
\frac{12(4x - 3)}{15} + \frac{10(6x - 3)}{15} = \frac{48x - 36}{15} + \frac{60x - 30}{15} = \frac{48x + 60x - 36 - 30}{15} = \frac{108x - 66}{15} = \frac{6(18x - 11)}{15}
\]

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Would you like a more detailed explanation for any of these steps or a different approach?

Here are some related questions:
1. How do you handle factoring expressions?
2. What is the process for simplifying expressions involving fractions?
3. Can you explain how to combine like terms in polynomial expressions?
4. What are common mistakes in distributing negative signs in expressions?
5. How do you approach solving algebraic equations after expansion?

**Tip**: When simplifying expressions, always look for like terms that can be combined after expanding, and be careful with distributing negative signs!

Expand the brackets and simplify these algebraic expressions.

This lesson covers expanding algebraic expressions with fractions and simplifying them step by step. It involves key concepts like the distributive property and combining like terms. Suitable for students in Grades 8-10, this problem helps strengthen algebraic manipulation skills.

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