Question

divide. \\(\dfrac{x + 3}{3x - 6} \div \dfrac{7x + 21}{6x - 12}\\) simplify your answer as much as possible.

Explanation:

Step1: Rewrite division as multiplication

To divide by a fraction, we multiply by its reciprocal. So, $\frac{x + 3}{3x - 6} \div \frac{7x + 21}{6x - 12}$ becomes $\frac{x + 3}{3x - 6} \times \frac{6x - 12}{7x + 21}$.

Step2: Factor the expressions

Factor the numerators and denominators:

• For the numerator $x + 3$, it remains as is.
• For the denominator $3x - 6$, factor out 3: $3(x - 2)$.
• For the numerator $6x - 12$, factor out 6: $6(x - 2)$.
• For the numerator $7x + 21$, factor out 7: $7(x + 3)$.

So the expression becomes $\frac{x + 3}{3(x - 2)} \times \frac{6(x - 2)}{7(x + 3)}$.

Step3: Cancel out common factors

We can cancel out the common factors $(x + 3)$ and $(x - 2)$ from the numerators and denominators:

• Cancel $(x + 3)$: $\frac{\cancel{x + 3}}{3(x - 2)} \times \frac{6(x - 2)}{7\cancel{(x + 3)}}$.
• Cancel $(x - 2)$: $\frac{\cancel{x + 3}}{3\cancel{(x - 2)}} \times \frac{6\cancel{(x - 2)}}{7\cancel{(x + 3)}}$.

After canceling, we have $\frac{6}{3\times7}$.

Step4: Simplify the remaining fraction

Simplify $\frac{6}{21}$ by dividing both the numerator and denominator by their greatest common divisor, which is 3. So, $\frac{6\div3}{21\div3}=\frac{2}{7}$.

Answer:

$\frac{2}{7}$