If $\frac{2}{2 + \frac{2}{2 + \frac{2}{x}}} = 3$, then $x$ is equal to

### Concept & Strategy Unlike standard continued fraction problems where you simplify from the bottom up, this is an **Inverse Continued Fraction Equation**. Because the unknown $x$ is buried at the very bottom, but the total value of the expression is known (it equals 3), the fastest strategy is to solve from the **top down**. ### Step-by-Step Solution We will isolate the terms layer by layer, starting from the outermost shell. * **Step 1: Isolate the main denominator** The equation is $\frac{2}{\text{Denominator}} = 3$. Rearranging this gives: $$\text{Denominator} = \frac{2}{3}$$ So, $2 + \frac{2}{2 + \frac{2}{x}} = \frac{2}{3}$ * **Step 2: Isolate the next nested fraction** Subtract 2 from both sides: $$\frac{2}{2 + \frac{2}{x}} = \frac{2}{3} - 2$$ $$\frac{2}{2 + \frac{2}{x}} = \frac{2 - 6}{3} = -\frac{4}{3}$$ * **Step 3: Isolate the next denominator** We have $\frac{2}{\text{Denominator}} = -\frac{4}{3}$. Rearranging gives $\text{Denominator} = 2 \times \left(-\frac{3}{4}\right) = -\frac{6}{4} = -\frac{3}{2}$. So, $2 + \frac{2}{x} = -\frac{3}{2}$ * **Step 4: Solve for $x$** Subtract 2 from both sides: $$\frac{2}{x} = -\frac{3}{2} - 2$$ $$\frac{2}{x} = \frac{-3 - 4}{2} = -\frac{7}{2}$$ Invert both sides and multiply to find $x$: $$\frac{x}{2} = -\frac{2}{7}$$ $$x = 2 \times \left(-\frac{2}{7}\right) = -\frac{4}{7}$$ ### Exam Strategy & Shortcut The top-down approach *is* the shortcut. If you try to simplify this from the bottom up (by finding a common denominator for $2 + 2/x$, then inverting, then adding 2, etc.), you will create massive, algebraically heavy expressions that are incredibly prone to silly mistakes. Working backwards with simple integers is always faster. ### Common Pitfall The biggest danger here is losing track of negative signs. In Step 2, subtracting 2 from $2/3$ yields a negative number ($-4/3$). If you accidentally write positive $4/3$, you will end up with an incorrect positive fraction for $x$. Always double-check your sign arithmetic when moving terms across the equals sign. ### Final Answer Therefore, the correct answer is **-\frac{4}{7}**.