Question

fully simplify.
\\(\dfrac{\dfrac{x}{9} - \dfrac{1}{3}}{\dfrac{x - 4}{3} + \dfrac{1}{x}}\\)

Explanation:

Step1: Simplify numerator and denominator separately.

First, simplify the numerator: $\frac{x}{9} - \frac{1}{3}$. Find a common denominator, which is 9. So $\frac{x}{9} - \frac{3}{9} = \frac{x - 3}{9}$.

Next, simplify the denominator: $\frac{x - 4}{3} + \frac{1}{x}$. Find a common denominator, which is $3x$. So $\frac{x(x - 4)}{3x} + \frac{3}{3x} = \frac{x(x - 4) + 3}{3x}$. Expand the numerator: $x^2 - 4x + 3$. Factor it: $(x - 1)(x - 3)$.

Step2: Rewrite the complex fraction.

The original expression becomes $\frac{\frac{x - 3}{9}}{\frac{(x - 1)(x - 3)}{3x}}$.

Step3: Divide the fractions.

Dividing by a fraction is multiplying by its reciprocal: $\frac{x - 3}{9} \times \frac{3x}{(x - 1)(x - 3)}$.

Step4: Cancel common factors.

Cancel out $(x - 3)$ from numerator and denominator, and simplify $\frac{3x}{9}$ to $\frac{x}{3}$. So we get $\frac{x}{3(x - 1)}$.

Answer:

$\frac{x}{3(x - 1)}$