To determine the product of the given rational expressions and simplify it to its reduced form, we will proceed step by step:
The given rational expressions are:
[tex]\[
\frac{x-3}{x+7} \cdot \frac{2x}{x-3}
\][/tex]
Step 1: Multiply the numerators and the denominators separately:
[tex]\[
\left( \frac{x-3}{x+7} \right) \cdot \left( \frac{2x}{x-3} \right) = \frac{(x-3) \cdot 2x}{(x+7) \cdot (x-3)}
\][/tex]
Step 2: Simplify the expression by canceling out common factors in the numerator and the denominator.
We notice that [tex]\((x-3)\)[/tex] appears in both the numerator and the denominator, so we can cancel this term:
[tex]\[
\frac{(x-3) \cdot 2x}{(x+7) \cdot (x-3)} = \frac{2x}{x+7}
\][/tex]
Step 3: The simplified expression is:
[tex]\[
\frac{2x}{x+7}
\][/tex]
Therefore, the product of the rational expressions in reduced form is:
[tex]\[
\frac{2x}{x+7}
\][/tex]
Comparing this result to the given choices:
A. [tex]\(\frac{2}{x-3}\)[/tex]
B. [tex]\(\frac{2x}{x+7}\)[/tex]
C. [tex]\(\frac{2}{x+7}\)[/tex]
D. [tex]\(\frac{2x}{x-3}\)[/tex]
The correct answer is:
B. [tex]\(\frac{2 x}{x+7}\)[/tex]
