Question

How can Ari simplify the following expression?

StartFraction 5 Over a minus 3 EndFraction minus 4 divided by 2 + StartFraction 1 Over a minus 3 EndFraction
Write the numerator and denominator with a common denominator. Then divide the numerator by the denominator. To do this, multiply the numerator by the reciprocal of the denominator.
Write the numerator and denominator with a common denominator. Then divide the numerator by the denominator. To do this, multiply the numerators and multiply the denominators.
Divide the numerator and the denominator by a – 3. Then divide the numerator by the denominator.
Divide the numerator and the denominator by a – 3. Then simplify the numerator and simplify the denominator.

Answer 1

The best choice is the first one.

Work:
4 ÷ (2 + 1/(a−3)) — write 2 + 1/(a−3) with common denominator (a−3):
2 + 1/(a−3) = (2a−5)/(a−3).
So 4 ÷ (2 + 1/(a−3)) = 4 ÷ ((2a−5)/(a−3)) = 4·(a−3)/(2a−5).

Then
5/(a−3) − 4·(a−3)/(2a−5) = [5(2a−5) − 4(a−3)^2] / [(a−3)(2a−5)]
= (−4a^2 + 34a − 61) / [(a−3)(2a−5)].

Domain: a ≠ 3, 5/2.