To divide the rational expressions
\[ \frac{y^2 - 4}{y - 2} \div \frac{5y - 10}{1} \]
we first rewrite the division as multiplication by the reciprocal:
\[ \frac{y^2 - 4}{y - 2} \times \frac{1}{5y - 10} \]
Next, we can factor both the numerator \(y^2 - 4\) and the denominator \(5y - 10\):
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\(y^2 - 4\) is a difference of squares: \[ y^2 - 4 = (y - 2)(y + 2) \]
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\(5y - 10\) can be factored out by 5: \[ 5y - 10 = 5(y - 2) \]
Now we substitute back into the expression:
\[ \frac{(y - 2)(y + 2)}{y - 2} \times \frac{1}{5(y - 2)} \]
Now we can simplify:
- Cancel \(y - 2\) in the numerator and denominator (as long as \(y \neq 2\)): \[ = \frac{y + 2}{5(y - 2)} \]
Now we have our simplified expression:
\[ \frac{y + 2}{5(y - 2)} \]
The restrictions on the variable \(y\) are that \(y \neq 2\) (from the original denominator) and that \(y - 2 \neq 0\).
Thus, the final answer is:
\[ \text{Answer: } \frac{y + 2}{5(y - 2)}, \text{ with restrictions } y \neq 2 \]
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