To simplify the expression \( \frac{v}{v-5} + \frac{9}{2v-10} \), we first observe that \( 2v - 10 \) can be factored:
\[ 2v - 10 = 2(v - 5) \]
Thus, we can rewrite the second term:
\[ \frac{9}{2v - 10} = \frac{9}{2(v - 5)} \]
Now we can express the two fractions with a common denominator. The common denominator is \( 2(v - 5) \). We rewrite the first fraction to have this denominator:
\[ \frac{v}{v - 5} = \frac{v \cdot 2}{(v - 5) \cdot 2} = \frac{2v}{2(v - 5)} \]
Now we can combine the two fractions:
\[ \frac{2v}{2(v - 5)} + \frac{9}{2(v - 5)} \]
Combine the numerators:
\[ \frac{2v + 9}{2(v - 5)} \]
This is the simplified expression. Now, we can check if the numerator \( 2v + 9 \) can be factored further, but it cannot.
Therefore, the simplified expression is:
\[ \frac{2v + 9}{2(v - 5)} \]
This is the final answer in its simplest form.
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