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Answer :
7. The simplified form of (1 / (1 + x/y)) is y / (y + x).
8. The simplified form of (3/(x - 4) - 4/(1 - 2/(x - 4))) is (-x - 2) / (x - 4)(x - 6).
7. To simplify (1 / (1 + x/y)), we can multiply the numerator and denominator by y to eliminate the fraction in the denominator:
(1 / (1 + x/y)) * (y/y) = y / (y + x).
Therefore, the simplified form of (1 / (1 + x/y)) is y / (y + x).
8. To simplify (3/(x - 4) - 4/(1 - 2/(x - 4))), we can simplify the second term by finding a common denominator:
4/(1 - 2/(x - 4)) = 4 / (1 - (2 / (x - 4))) * ((x - 4) / (x - 4)) = 4(x - 4) / (x - 4 - 2) = 4(x - 4) / (x - 6).
Now, we can combine the two terms with a common denominator:
3/(x - 4) - 4/(1 - 2/(x - 4)) = 3/(x - 4) - 4(x - 4) / (x - 6).
To simplify further, we can find a common denominator for the two terms:
3/(x - 4) - 4(x - 4) / (x - 6) = (3(x - 6) - 4(x - 4)) / (x - 4)(x - 6).
Simplifying the numerator:
(3x - 18 - 4x + 16) / (x - 4)(x - 6) = (-x - 2) / (x - 4)(x - 6).
Therefore, the simplified form of (3/(x - 4) - 4/(1 - 2/(x - 4))) is (-x - 2) / (x - 4)(x - 6).
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