https://file.aiquickdraw.com/mathbot/file/1728585399570xm85cu7q.jpg

Let's solve the multi-step equations with fractions from the image.

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### a) \( 6x \left( \frac{1}{2} + \frac{1}{3} \right) = 2x + 3 \)

1. First, simplify the terms inside the parentheses:
   \[
   \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}
   \]
   So the equation becomes:
   \[
   6x \cdot \frac{5}{6} = 2x + 3
   \]
2. Simplify \( 6x \cdot \frac{5}{6} \):
   \[
   x \cdot 5 = 5x
   \]
   Now the equation is:
   \[
   5x = 2x + 3
   \]
3. Subtract \( 2x \) from both sides:
   \[
   5x - 2x = 3
   \]
   \[
   3x = 3
   \]
4. Solve for \( x \):
   \[
   x = 1
   \]

---

### b) \( \frac{2}{3}(x + 5) = \frac{4}{9} \)

1. Distribute \( \frac{2}{3} \) over \( (x + 5) \):
   \[
   \frac{2}{3}x + \frac{2}{3} \cdot 5 = \frac{2}{3}x + \frac{10}{3}
   \]
   So the equation becomes:
   \[
   \frac{2}{3}x + \frac{10}{3} = \frac{4}{9}
   \]
2. To eliminate the fractions, multiply the entire equation by 9:
   \[
   9 \cdot \left( \frac{2}{3}x + \frac{10}{3} \right) = 9 \cdot \frac{4}{9}
   \]
   \[
   6x + 30 = 4
   \]
3. Subtract 30 from both sides:
   \[
   6x = 4 - 30
   \]
   \[
   6x = -26
   \]
4. Solve for \( x \):
   \[
   x = \frac{-26}{6} = \frac{-13}{3}
   \]

---

### c) \( \frac{1}{5} \left( \frac{x}{2} + \frac{3}{4} \right) = \frac{1}{4} \)

1. First, simplify the terms inside the parentheses:
   \[
   \frac{x}{2} + \frac{3}{4} = \frac{2x}{4} + \frac{3}{4} = \frac{2x + 3}{4}
   \]
   So the equation becomes:
   \[
   \frac{1}{5} \cdot \frac{2x + 3}{4} = \frac{1}{4}
   \]
2. Multiply both sides by 5 to eliminate the \( \frac{1}{5} \):
   \[
   \frac{2x + 3}{4} = \frac{5}{4}
   \]
3. Multiply both sides by 4 to eliminate the denominator:
   \[
   2x + 3 = 5
   \]
4. Subtract 3 from both sides:
   \[
   2x = 5 - 3 = 2
   \]
5. Solve for \( x \):
   \[
   x = \frac{2}{2} = 1
   \]

---

### d) \( \frac{x + 3}{2} + (x - 1) = \frac{4}{5} \)

1. Multiply the entire equation by 10 to eliminate the fractions:
   \[
   10 \cdot \left( \frac{x + 3}{2} \right) + 10 \cdot (x - 1) = 10 \cdot \frac{4}{5}
   \]
   \[
   5(x + 3) + 10(x - 1) = 8
   \]
2. Distribute the terms:
   \[
   5x + 15 + 10x - 10 = 8
   \]
   Simplify:
   \[
   15x + 5 = 8
   \]
3. Subtract 5 from both sides:
   \[
   15x = 3
   \]
4. Solve for \( x \):
   \[
   x = \frac{3}{15} = \frac{1}{5}
   \]

---

Would you like further explanations on any step? Here are some related questions for practice:

1. How would you simplify equations that have variables on both sides?
2. What is the process of eliminating fractions in multi-step equations?
3. How can you check your solution for fractional equations?
4. How do you factor equations that involve fractions?
5. How would you approach solving an equation if all terms are fractions?

**Tip**: Always try to eliminate fractions early by multiplying through by the least common denominator.

Solve multi-step equations with fractions as shown in the image.

Learn to solve multi-step linear equations with fractions. This worksheet covers fundamental algebra concepts, including simplifying fractions and using properties of equality. Ideal for Grades 8-10 students.

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