https://file.aiquickdraw.com/mathbot/file/1726102899953nps989zh.jpg

Let's break down and solve each of the complex fraction problems from the image step by step.

### 1. \( 10 = \dfrac{1}{1 + \dfrac{1}{1 + \dfrac{1}{x}}} \)

First, we start from the innermost fraction and work outward:

\[
1 + \dfrac{1}{x} = \dfrac{x + 1}{x}
\]

Now substitute this back into the next part of the expression:

\[
1 + \dfrac{1}{\dfrac{x + 1}{x}} = 1 + \dfrac{x}{x + 1} = \dfrac{x + 1 + x}{x + 1} = \dfrac{2x + 1}{x + 1}
\]

Finally, substitute into the outermost fraction:

\[
\dfrac{1}{1 + \dfrac{1}{x + 1}} = \dfrac{x + 1}{2x + 1}
\]

Now set this equal to 10:

\[
10 = \dfrac{x + 1}{2x + 1}
\]

Multiply both sides by \( 2x + 1 \):

\[
10(2x + 1) = x + 1
\]

Expand:

\[
20x + 10 = x + 1
\]

Now subtract \( x \) and 10 from both sides:

\[
19x = -9
\]

Solve for \( x \):

\[
x = \dfrac{-9}{19}
\]

---

### 2. \( \dfrac{\dfrac{x}{1}}{1 - \dfrac{1}{2x}} = 1 \)

Simplify the denominator:

\[
1 - \dfrac{1}{2x} = \dfrac{2x - 1}{2x}
\]

Now rewrite the expression:

\[
\dfrac{x}{\dfrac{2x - 1}{2x}} = 1
\]

Multiply by the reciprocal:

\[
x \times \dfrac{2x}{2x - 1} = 1
\]

This simplifies to:

\[
\dfrac{2x^2}{2x - 1} = 1
\]

Multiply both sides by \( 2x - 1 \):

\[
2x^2 = 2x - 1
\]

Rearrange into a quadratic equation:

\[
2x^2 - 2x + 1 = 0
\]

Now solve using the quadratic formula:

\[
x = \dfrac{-(-2) \pm \sqrt{(-2)^2 - 4(2)(1)}}{2(2)} = \dfrac{2 \pm \sqrt{4 - 8}}{4} = \dfrac{2 \pm \sqrt{-4}}{4}
\]

This gives:

\[
x = \dfrac{2 \pm 2i}{4} = \dfrac{1 \pm i}{2}
\]

Thus, \( x = \dfrac{1 + i}{2} \) or \( x = \dfrac{1 - i}{2} \).

---

### 3. \( \dfrac{\dfrac{1}{2} + \dfrac{1}{3}}{\dfrac{1}{4} + \dfrac{1}{5}} = \dfrac{1}{x} \)

First, simplify the numerator and the denominator separately:

\[
\dfrac{1}{2} + \dfrac{1}{3} = \dfrac{3 + 2}{6} = \dfrac{5}{6}
\]

\[
\dfrac{1}{4} + \dfrac{1}{5} = \dfrac{5 + 4}{20} = \dfrac{9}{20}
\]

Now, the equation becomes:

\[
\dfrac{\dfrac{5}{6}}{\dfrac{9}{20}} = \dfrac{1}{x}
\]

Simplify:

\[
\dfrac{5}{6} \times \dfrac{20}{9} = \dfrac{100}{54} = \dfrac{50}{27}
\]

Thus, the equation becomes:

\[
\dfrac{50}{27} = \dfrac{1}{x}
\]

Solve for \( x \):

\[
x = \dfrac{27}{50}
\]

---

### 4. \( \d

Solve the following complex fraction problems.

This webpage provides step-by-step solutions to complex fraction problems involving algebraic manipulation and solving quadratic equations. Topics covered include simplification of complex fractions and the use of the quadratic formula.

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