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Let's go through each problem and solve the rational equations step by step.

### 1) \(\frac{1}{3} + \frac{4}{5} = \frac{x}{9}\)

**Solution:**
1. Combine the fractions on the left-hand side:
   \[
   \frac{1}{3} + \frac{4}{5} = \frac{5 + 12}{15} = \frac{17}{15}
   \]

2. Now, equate it to \(\frac{x}{9}\):
   \[
   \frac{17}{15} = \frac{x}{9}
   \]

3. Cross-multiply:
   \[
   17 \times 9 = 15 \times x
   \]
   \[
   153 = 15x
   \]

4. Solve for \(x\):
   \[
   x = \frac{153}{15} = 10.2
   \]

Thus, the solution for the first equation is \(x = 10.2\).

---

### 2) \(\frac{1}{3} - \frac{1}{x} = \frac{5}{6}\)

**Solution:**
1. Rearrange the equation:
   \[
   \frac{1}{3} - \frac{5}{6} = \frac{1}{x}
   \]

2. Find a common denominator for the left-hand side:
   \[
   \frac{1}{3} = \frac{2}{6}
   \]
   \[
   \frac{2}{6} - \frac{5}{6} = \frac{-3}{6} = -\frac{1}{2}
   \]

3. So, the equation becomes:
   \[
   \frac{1}{x} = -\frac{1}{2}
   \]

4. Invert both sides to solve for \(x\):
   \[
   x = -2
   \]

Thus, the solution for the second equation is \(x = -2\).

---

### 3) \(\frac{2}{3} + \frac{1}{5} = \frac{7}{3x}\)

**Solution:**
1. Combine the fractions on the left-hand side:
   \[
   \frac{2}{3} + \frac{1}{5} = \frac{10 + 3}{15} = \frac{13}{15}
   \]

2. Now, equate it to \(\frac{7}{3x}\):
   \[
   \frac{13}{15} = \frac{7}{3x}
   \]

3. Cross-multiply:
   \[
   13 \times 3x = 15 \times 7
   \]
   \[
   39x = 105
   \]

4. Solve for \(x\):
   \[
   x = \frac{105}{39} = \frac{35}{13} \approx 2.69
   \]

Thus, the solution for the third equation is \(x \approx 2.69\).

---

Would you like any additional details on these solutions? Here are 5 related questions you might find useful:

1. What are common methods to combine fractions with different denominators?
2. How does cross-multiplying work in rational equations?
3. Can you explain how to check solutions to rational equations for correctness?
4. What happens if the denominator in a rational equation equals zero?
5. How do you deal with negative solutions in rational equations?

**Tip:** Always verify solutions by substituting them back into the original equation to ensure accuracy!

Step-by-step solutions for solving rational equations involving fractions, including cross-multiplication and combining fractions. Learn to solve equations like 1/3 + 4/5 = x/9. Suitable for students in Grades 8-10.

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