Question

module 7 pretest: rational expressions and equations
module 7 pretest: rational expressions and equations
score: 2/9 answered: 2/9
question 3
divide and simplify:
\\(\frac{x^2 - 81}{x^2 - 5x - 36} \div \frac{2x}{x + 4}\\)

Explanation:

Step1: Rewrite division as multiplication

To divide rational expressions, we multiply by the reciprocal of the divisor. So, \(\frac{x^2 - 81}{x^2 - 5x - 36} \div \frac{2x}{x + 4}\) becomes \(\frac{x^2 - 81}{x^2 - 5x - 36} \times \frac{x + 4}{2x}\).

Step2: Factor numerators and denominators

• Factor \(x^2 - 81\) (difference of squares): \(x^2 - 81=(x - 9)(x + 9)\)
• Factor \(x^2 - 5x - 36\): We need two numbers that multiply to \(-36\) and add to \(-5\). Those numbers are \(-9\) and \(4\), so \(x^2 - 5x - 36=(x - 9)(x + 4)\)

Substituting these factorizations into the expression, we get: \(\frac{(x - 9)(x + 9)}{(x - 9)(x + 4)} \times \frac{x + 4}{2x}\)

Step3: Cancel common factors

• Cancel \((x - 9)\) from the numerator and denominator.
• Cancel \((x + 4)\) from the numerator and denominator.

After canceling, we are left with \(\frac{x + 9}{2x}\)

Answer:

\(\frac{x + 9}{2x}\)