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I have opened the image you uploaded, which contains several mathematical problems involving fractions. Let's go through each problem step by step.

### Problem 1
\[
10 = \frac{1}{1 + \frac{1}{1 + \frac{1}{x}}}
\]
We will simplify the fraction step by step to solve for \( x \).

1. Combine the innermost fraction first: \( \frac{1}{x} \).
2. Add 1 to this fraction: \( 1 + \frac{1}{x} = \frac{x+1}{x} \).
3. In the next step, simplify \( 1 + \frac{1}{1 + \frac{1}{x}} \) by computing \( 1 + \frac{1}{\frac{x+1}{x}} = \frac{x+2}{x+1} \).
4. Now, \( \frac{1}{1 + \frac{1}{1 + \frac{1}{x}}} = \frac{x+1}{x+2} \).
5. We equate this to 10:
   \[
   10 = \frac{x+1}{x+2}
   \]
6. Solving \( 10(x+2) = x+1 \) gives:
   \[
   10x + 20 = x + 1 \quad \Rightarrow \quad 9x = -19 \quad \Rightarrow \quad x = -\frac{19}{9}
   \]

Thus, \( x = -\frac{19}{9} \).

---

### Problem 2
\[
\frac{\frac{x}{1}}{\frac{1}{2x} - 3} = 1
\]
1. Simplify \( \frac{x}{1} = x \), so:
   \[
   \frac{x}{\frac{1}{2x} - 3} = 1
   \]
2. Simplify the denominator:
   \[
   \frac{1}{2x} - 3 = \frac{1 - 6x}{2x}
   \]
3. Now solve:
   \[
   \frac{x}{\frac{1-6x}{2x}} = 1 \quad \Rightarrow \quad x \cdot \frac{2x}{1-6x} = 1
   \]
4. Simplify and solve:
   \[
   \frac{2x^2}{1-6x} = 1 \quad \Rightarrow \quad 2x^2 = 1 - 6x
   \]
   Rearranging gives the quadratic equation:
   \[
   2x^2 + 6x - 1 = 0
   \]
5. Use the quadratic formula:
   \[
   x = \frac{-6 \pm \sqrt{6^2 - 4(2)(-1)}}{2(2)} = \frac{-6 \pm \sqrt{36 + 8}}{4} = \frac{-6 \pm \sqrt{44}}{4} = \frac{-6 \pm 2\sqrt{11}}{4}
   \]
   Simplifying:
   \[
   x = \frac{-3 \pm \sqrt{11}}{2}
   \]

So the solution is \( x = \frac{-3 \pm \sqrt{11}}{2} \).

---

Would you like the solutions for the remaining problems, or need clarifications on these? Here are five additional questions based on what we've covered:

1. How does simplifying complex fractions help in solving for \( x \)?
2. What is the quadratic formula and when is it used?
3. How do we handle nested fractions?
4. Can you explain why adding fractions with different denominators requires finding a common denominator?
5. What strategies are useful for solving equations involving fractions?

**Tip:** Always simplify fractions step by step to avoid mistakes, especially when dealing with nested expressions.

Simplify the given complex fractions and solve for the value of x.

Learn how to simplify complex fractions and solve quadratic equations through a step-by-step process. This problem involves core algebraic concepts such as handling nested fractions and applying the quadratic formula. Suitable for students in Grades 9-11.

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