Question

divide.
\\(\frac{9x - 27}{18x + 4} \div \frac{3x - 9}{9x + 2}\\)
simplify your answer as much as possible.

Explanation:

Step1: Rewrite division as multiplication

To divide two fractions, we multiply the first fraction by the reciprocal of the second fraction. So, \(\frac{9x - 27}{18x + 4}\div\frac{3x - 9}{9x + 2}=\frac{9x - 27}{18x + 4}\times\frac{9x + 2}{3x - 9}\)

Step2: Factor numerators and denominators

Factor out the greatest common factor (GCF) from each polynomial:

• For \(9x - 27\), the GCF of \(9x\) and \(-27\) is \(9\), so \(9x - 27 = 9(x - 3)\)
• For \(3x - 9\), the GCF of \(3x\) and \(-9\) is \(3\), so \(3x - 9 = 3(x - 3)\)
• The denominator \(18x + 4\) has a GCF of \(2\), so \(18x + 4 = 2(9x + 2)\)

Substituting these factored forms into the expression, we get:
\(\frac{9(x - 3)}{2(9x + 2)}\times\frac{9x + 2}{3(x - 3)}\)

Step3: Cancel common factors

We can cancel out the common factors \((x - 3)\) and \((9x + 2)\) (assuming \(x
eq3\) and \(x
eq-\frac{2}{9}\) to avoid division by zero):
\(\frac{9\cancel{(x - 3)}}{2\cancel{(9x + 2)}}\times\frac{\cancel{9x + 2}}{3\cancel{(x - 3)}}=\frac{9}{2}\times\frac{1}{3}\)

Step4: Multiply the remaining factors

Multiply the numerators and denominators: \(\frac{9\times1}{2\times3}=\frac{9}{6}\)

Step5: Simplify the fraction

Simplify \(\frac{9}{6}\) by dividing both the numerator and denominator by their GCF, which is \(3\): \(\frac{9\div3}{6\div3}=\frac{3}{2}\)

Answer:

\(\frac{3}{2}\)