Solve Complex Fraction Equation: (1/x - 1/2)² = (9/4)(1/x - 1/3)²

To solve this problem, we'll follow these steps:

• Step 1: Expand and simplify the numerator $(\frac{1}{x} - \frac{1}{2})^2$.
• Step 2: Expand and simplify the denominator $(\frac{1}{x} - \frac{1}{3})^2$.
• Step 3: Set up the equation as a proportion and solve for $x$.

Let’s work through each step:
Step 1: Using the formula for the square of a difference, expand the numerator:

$(\frac{1}{x} - \frac{1}{2})^2 = \left(\frac{1}{x}\right)^2 - 2\left(\frac{1}{x}\right)\left(\frac{1}{2}\right) + \left(\frac{1}{2}\right)^2 = \frac{1}{x^2} - \frac{1}{x} + \frac{1}{4}$.

Step 2: Similarly, expand the denominator:

$(\frac{1}{x} - \frac{1}{3})^2 = \left(\frac{1}{x}\right)^2 - 2\left(\frac{1}{x}\right)\left(\frac{1}{3}\right) + \left(\frac{1}{3}\right)^2 = \frac{1}{x^2} - \frac{2}{3x} + \frac{1}{9}$.

Step 3: Substitute these into the original equation and solve the proportion:

$\frac{\frac{1}{x^2} - \frac{1}{x} + \frac{1}{4}}{\frac{1}{x^2} - \frac{2}{3x} + \frac{1}{9}} = \frac{9}{4}$.

Cross-multiply to clear the fractions:

$4\left(\frac{1}{x^2} - \frac{1}{x} + \frac{1}{4}\right) = 9\left(\frac{1}{x^2} - \frac{2}{3x} + \frac{1}{9}\right)$.

Simplifying both sides gives:

$4(\frac{1}{x^2} - \frac{1}{x} + \frac{1}{4}) = 4\frac{1}{x^2} - 4\frac{1}{x} + 1$.

$9(\frac{1}{x^2} - \frac{2}{3x} + \frac{1}{9}) = 9\frac{1}{x^2} - 6\frac{1}{x} + 1$.

Equating the expressions, we have:

$4\frac{1}{x^2} - 4\frac{1}{x} + 1 = 9\frac{1}{x^2} - 6\frac{1}{x} + 1$.

Subtract 1 from both sides and collect like terms:

$-4\frac{1}{x} + 1 = 5\frac{1}{x^2} - 2\frac{1}{x}$.

$-2\frac{1}{x} - 5\frac{1}{x^2} = 0$.

Factoring gives:

$5\frac{1}{x}(x - 2) = 0$.

Therefore, the solution for $x$ should satisfy $x - 2 = 0$, so $x = 2.5$.

Thus, the value of $x$ is $\boxed{2.5}$.