Question

select the correct answer. which rational expression is equivalent to this expression? \boxed{\dfrac{4}{x - 3}} a. \dfrac{x - 3}{x + 2} \div \dfrac{4}{x + 2} b. \dfrac{x + 2}{x - 3} \div \dfrac{4}{x + 2} c. \dfrac{x - 3}{x + 2} \cdot \dfrac{x + 2}{4} d. \dfrac{x + 2}{x - 3} \cdot \dfrac{4}{x + 2}

Explanation:

Step1: Recall fraction division rule

Dividing by a fraction = multiply by its reciprocal: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

Step2: Simplify Option A

Cancel common terms:
$\frac{x-3}{x+2} \div \frac{4}{x+2} = \frac{x-3}{x+2} \times \frac{x+2}{4} = \frac{x-3}{4}$

Step3: Simplify Option B

Apply division rule, multiply terms:
$\frac{x+2}{x-3} \div \frac{4}{x+2} = \frac{x+2}{x-3} \times \frac{x+2}{4} = \frac{(x+2)^2}{4(x-3)}$

Step4: Simplify Option C

Multiply numerators and denominators:
$\frac{x-3}{x+2} \cdot \frac{x+2}{4} = \frac{x-3}{4}$

Step5: Simplify Option D

Cancel common terms:
$\frac{x+2}{x-3} \cdot \frac{4}{x+2} = \frac{4}{x-3}$

Answer:

D. $\frac{x+2}{x-3} \cdot \frac{4}{x+2}$