To simplify the expression \( \frac{10v - 8}{v + 10} - \frac{2}{v + 1} \), we first need to find a common denominator. The denominators here are \( v + 10 \) and \( v + 1 \). The common denominator will be \( (v + 10)(v + 1) \).
Now we can rewrite each fraction with the common denominator:
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For the first fraction: \[ \frac{10v - 8}{v + 10} = \frac{(10v - 8)(v + 1)}{(v + 10)(v + 1)} \]
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For the second fraction: \[ \frac{2}{v + 1} = \frac{2(v + 10)}{(v + 1)(v + 10)} \]
Now, we can combine the two fractions: \[ \frac{(10v - 8)(v + 1) - 2(v + 10)}{(v + 10)(v + 1)} \]
Next, we simplify the numerator. First, expand both terms in the numerator:
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Expanding \( (10v - 8)(v + 1) \): \[ 10v(v) + 10v(1) - 8(v) - 8(1) = 10v^2 + 10v - 8v - 8 = 10v^2 + 2v - 8 \]
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Expanding \( -2(v + 10) \): \[ -2v - 20 \]
Now combine these two results: \[ 10v^2 + 2v - 8 - 2v - 20 \] Combine like terms: \[ 10v^2 - 28 \]
Thus, the numerator becomes: \[ 10v^2 - 28 \]
Now our expression is: \[ \frac{10v^2 - 28}{(v + 10)(v + 1)} \]
Next, we can factor the numerator: \[ 10(v^2 - 2.8) = 10(v^2 - 7) \]
Thus, our expression is: \[ \frac{10(v^2 - 7)}{(v + 10)(v + 1)} \]
Finally, since \( v^2 - 7 \) doesn’t have any factors that can be canceled with the denominator, this is our simplified answer.
The final expression in simplest form is: \[ \frac{10(v^2 - 7)}{(v + 10)(v + 1)} \]