To find the product of the given rational expressions:
[tex]\[
\frac{2}{x} \cdot \frac{3}{2x - 5}
\][/tex]
we need to follow these steps:
1. Multiply the numerators:
Identify the numerators of each fraction and multiply them together. In this case, the numerators are 2 and 3.
[tex]\[
2 \times 3 = 6
\][/tex]
2. Multiply the denominators:
Similarly, identify the denominators of each fraction and multiply them. The denominators are [tex]\( x \)[/tex] and [tex]\( 2x - 5 \)[/tex].
[tex]\[
x \times (2x - 5) = x(2x - 5)
\][/tex]
3. Simplify the product of the denominators:
Now, distribute [tex]\( x \)[/tex] in the denominator expression.
[tex]\[
x \times (2x - 5) = 2x^2 - 5x
\][/tex]
4. Form the product of the fractions:
Combine the results from the steps above. The product of the rational expressions is the fraction formed by these new numerator and denominator.
[tex]\[
\frac{6}{2x^2 - 5x}
\][/tex]
Now, compare this result with the given options:
A. [tex]\(\frac{6}{2x^2 - 5x}\)[/tex] \
B. [tex]\(\frac{3}{x^2 - 5x}\)[/tex] \
C. [tex]\(\frac{6}{2x - 5}\)[/tex] \
D. [tex]\(\frac{6x}{x - 5}\)[/tex]
Clearly, the product we derived matches option A:
[tex]\[
\frac{6}{2x^2 - 5x}
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{\frac{6}{2x^2 - 5x}}
\][/tex]
This corresponds to option A.
