https://file.aiquickdraw.com/mathbot/file/1726100384679zyt9j9ou.jpg

Let's break down each of the five equations from the image and solve for \( x \) one by one.

### 1. Equation:
\[
10 = \frac{1}{1 + \frac{1}{1 + \frac{1}{x}}}
\]
We will simplify this nested fraction step by step:
1. Start with the innermost fraction: \( \frac{1}{x} \).
2. Add 1 to it: \( 1 + \frac{1}{x} = \frac{x + 1}{x} \).
3. Invert and add 1 again: \( 1 + \frac{1}{\frac{x+1}{x}} = \frac{2x + 1}{x} \).
4. Finally, invert and equate: \( \frac{1}{\frac{2x+1}{x}} = \frac{x}{2x + 1} \).

Thus, the equation becomes:
\[
10 = \frac{x}{2x + 1}
\]
Multiplying both sides by \( 2x + 1 \):
\[
10(2x + 1) = x \implies 20x + 10 = x \implies 19x = -10 \implies x = -\frac{10}{19}
\]

### 2. Equation:
\[
\frac{\frac{x}{1}}{\frac{1}{2x} - 3} = 1
\]
We simplify the denominator:
1. \( \frac{1}{2x} - 3 = \frac{1}{2x} - \frac{6x}{2x} = \frac{1 - 6x}{2x} \).

Now the equation becomes:
\[
\frac{x}{\frac{1 - 6x}{2x}} = 1
\]
Multiplying by the reciprocal of the denominator:
\[
x \cdot \frac{2x}{1 - 6x} = 1 \implies \frac{2x^2}{1 - 6x} = 1
\]
Multiplying both sides by \( 1 - 6x \):
\[
2x^2 = 1 - 6x
\]
Rearranging:
\[
2x^2 + 6x - 1 = 0
\]
We solve this quadratic equation using the quadratic formula:
\[
x = \frac{-6 \pm \sqrt{6^2 - 4 \cdot 2 \cdot (-1)}}{2 \cdot 2} = \frac{-6 \pm \sqrt{36 + 8}}{4} = \frac{-6 \pm \sqrt{44}}{4} = \frac{-6 \pm 2\sqrt{11}}{4} = \frac{-3 \pm \sqrt{11}}{2}
\]

### 3. Equation:
\[
\frac{\frac{1}{2} + \frac{1}{3}}{\frac{1}{4} + \frac{1}{5}} = \frac{1}{x}
\]
Simplify both the numerator and denominator:
1. \( \frac{1}{2} + \frac{1}{3} = \frac{3 + 2}{6} = \frac{5}{6} \).
2. \( \frac{1}{4} + \frac{1}{5} = \frac{5 + 4}{20} = \frac{9}{20} \).

Thus, the equation becomes:
\[
\frac{\frac{5}{6}}{\frac{9}{20}} = \frac{1}{x}
\]
Simplifying the fraction:
\[
\frac{5}{6} \cdot \frac{20}{9} = \frac{100}{54} = \frac{50}{27}
\]
So:
\[
\frac{50}{27} = \frac{1}{x} \implies x = \frac{27}{50}
\]

### 4. Equation:
\[
\frac{\frac{2}{2 + \frac{2}{3}}}{\frac{x}{3 + \frac{3}{x}}} = \frac{2}{3}
\]
Let's simplify each part:
1. \( 2 + \frac{2}{3} = \frac{6 + 2}{3} = \frac{8}{3} \), so \( \frac{2}{2 + \frac{2}{3}} = \frac{

Simplify each complex fraction and find the value of x.

Solve equations involving complex fractions to find the value of x. This problem set covers simplification of complex fractions, algebraic manipulation, and solving quadratic equations. Suitable for Grades 10-12.

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