To find the sum of the two rational expressions [tex]\(\frac{2 x - 1}{7 x} + \frac{x}{x - 2}\)[/tex], we need to follow a series of algebraic steps. Let's break them down in detail:
1. Identify the given expressions:
[tex]\[
\frac{2 x - 1}{7 x} \quad \text{and} \quad \frac{x}{x - 2}
\][/tex]
2. Find the common denominator:
The common denominator for the two expressions is the product of the individual denominators, which is [tex]\(7x(x - 2)\)[/tex].
3. Rewrite each fraction with the common denominator:
[tex]\[
\frac{2 x - 1}{7 x} = \frac{(2 x - 1) \cdot (x - 2)}{7 x (x - 2)} \quad \text{and} \quad \frac{x}{x - 2} = \frac{x \cdot 7 x}{7 x (x - 2)}
\][/tex]
4. Expand the numerators:
For the first expression,
[tex]\[
\frac{(2 x - 1) \cdot (x - 2)}{7 x (x - 2)} = \frac{2 x \cdot x - 2 x \cdot 2 - 1 \cdot x + 1 \cdot 2}{7 x (x - 2)} = \frac{2 x^2 - 4 x - x + 2}{7 x (x - 2)} = \frac{2 x^2 - 5 x + 2}{7 x (x - 2)}
\][/tex]
For the second expression,
[tex]\[
\frac{x \cdot 7 x}{7 x (x - 2)} = \frac{7 x^2}{7 x (x - 2)}
\][/tex]
5. Add the two expressions:
[tex]\[
\frac{2 x^2 - 5 x + 2}{7 x (x - 2)} + \frac{7 x^2}{7 x (x - 2)} = \frac{(2 x^2 - 5 x + 2) + 7 x^2}{7 x (x - 2)}
\][/tex]
6. Combine and simplify the numerators:
[tex]\[
(2 x^2 - 5 x + 2) + 7 x^2 = 9 x^2 - 5 x + 2
\][/tex]
So the combined expression is:
[tex]\[
\frac{9 x^2 - 5 x + 2}{7 x (x - 2)}
\][/tex]
7. Check the options:
Now, we compare our result with the given options:
[tex]\[
\text{A.} \frac{9 x^2 - 5 x + 2}{7 x^2 - 2}
\][/tex]
[tex]\[
\text{B.} \frac{2 x^2 - 5 x + 2}{7 x^2 - 14 x}
\][/tex]
[tex]\[
\text{C.} \frac{3 x - 1}{8 x - 2}
\][/tex]
[tex]\[
\text{D.} \frac{9 x^2 - 5 x + 2}{7 x^2 - 14 x}
\][/tex]
Our simplified result [tex]\(\frac{9 x^2 - 5 x + 2}{7 x (x - 2)}\)[/tex] matches the form of option D when rewritten, as 7x(x-2) is indeed [tex]\(7 x^2 - 14 x\)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{D}
\][/tex]
