Question

topic: complex rational expressions
progress:
the movement of the progress bar may be uneven because questions can be
simplify: $\frac{7-\frac{7}{3x}}{\frac{5}{x}-\frac{5}{3x^2}}$
$\frac{14x}{5}$
$\frac{7x}{5}$
$35x$
$\frac{21x^2-7x}{15x-5}$

Explanation:

Step1: Find LCD of numerator

The least common denominator (LCD) of the numerator $7 - \frac{7}{3x}$ is $3x$. Rewrite $7$ as a fraction with denominator $3x$:
$7 = \frac{21x}{3x}$, so the numerator becomes $\frac{21x}{3x} - \frac{7}{3x} = \frac{21x - 7}{3x}$

Step2: Find LCD of denominator

The LCD of the denominator $\frac{5}{x} - \frac{5}{3x^2}$ is $3x^2$. Rewrite $\frac{5}{x}$ as a fraction with denominator $3x^2$:
$\frac{5}{x} = \frac{15x}{3x^2}$, so the denominator becomes $\frac{15x}{3x^2} - \frac{5}{3x^2} = \frac{15x - 5}{3x^2}$

Step3: Rewrite as fraction division

The original expression is now the numerator divided by the denominator:
$\frac{\frac{21x - 7}{3x}}{\frac{15x - 5}{3x^2}}$
This is equivalent to multiplying the numerator by the reciprocal of the denominator:
$\frac{21x - 7}{3x} \times \frac{3x^2}{15x - 5}$

Step4: Factor and simplify

Factor numerator and denominator terms:
$21x - 7 = 7(3x - 1)$, $15x - 5 = 5(3x - 1)$
Substitute these into the expression:
$\frac{7(3x - 1)}{3x} \times \frac{3x^2}{5(3x - 1)}$
Cancel common factors $3x$, $(3x - 1)$:
$\frac{7}{1} \times \frac{x}{5} = \frac{7x}{5}$

Answer:

$\boldsymbol{\frac{7x}{5}}$ (corresponding to the second option)