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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.

The image contains several mathematical problems involving complex fractions and solving for x. For example, simplifying nested fractions and solving quadratic equations.

Learn how to solve complex fractions with nested expressions and quadratic equations. This page covers simplifying fractions and applying the quadratic formula, suitable for high school students.

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