To simplify the expression
\[ \frac{q}{10q + 5} - \frac{6}{30q + 15}, \]
we first notice that both denominators can be factored.
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Factoring the denominators:
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For \(10q + 5\): \[ 10q + 5 = 5(2q + 1) \]
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For \(30q + 15\): \[ 30q + 15 = 15(2q + 1) \]
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Thus, we can rewrite the expression as:
\[ \frac{q}{5(2q + 1)} - \frac{6}{15(2q + 1)}. \]
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Finding a common denominator:
The common denominator for the two fractions is \(15(2q + 1)\). We can rewrite the first fraction to have this common denominator:
\[ \frac{q}{5(2q + 1)} = \frac{3q}{15(2q + 1)}. \]
Now our expression can be written as:
\[ \frac{3q}{15(2q + 1)} - \frac{6}{15(2q + 1)}. \]
- Combining the fractions:
We can now combine the fractions over the common denominator:
\[ \frac{3q - 6}{15(2q + 1)}. \]
- Simplifying the numerator:
The numerator \(3q - 6\) can be factored:
\[ 3q - 6 = 3(q - 2). \]
Thus, we rewrite the fraction:
\[ \frac{3(q - 2)}{15(2q + 1)}. \]
- Simplifying the fraction:
Now we can cancel the common factor of \(3\) in the numerator and \(15\) in the denominator:
\[ \frac{3(q - 2)}{15(2q + 1)} = \frac{q - 2}{5(2q + 1)}. \]
Putting this all together, the simplified expression is:
\[ \frac{q - 2}{5(2q + 1)}. \]
Thus, the final answer is:
\[ \boxed{\frac{q - 2}{5(2q + 1)}}. \]
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