Answer
a. $\frac{x^2y}{y + x}$
b. $\frac{(x - 1)^2}{2x}$
Work Step by Step
a. Do the addition in the denominator first.
Find the least common denominator (LCD) of the two terms of the denominator. Then, multiply the numerator of each fraction by the factor that is missing between its denominator and the LCD:
$\frac{x}{\frac{1(y)}{xy} + \frac{1(x)}{xy}}$
Perform the addition in the denominator:
$\frac{x}{\frac{y + x}{xy}}$
Rewrite this fraction as a division problem:
$x \div \frac{y + x}{xy}$
To divide one fraction by another, multiply the first fraction by the reciprocal of the second fraction:
$x \bullet \frac{xy}{y + x}$
Multiply the numerators together, and multiply the denominators together:
$\frac{x^2y}{y + x}$
b. Find the least common denominator (LCD) of the two terms in the numerator and the two terms in the denominator. Then, multiply the numerator of each fraction by the factor that is missing between its denominator and the LCD:
$\frac{\frac{(x - 2)(x + 1)}{x(x + 1)} + \frac{2(x)}{x(x + 1)}}{\frac{3(x + 1)}{(x - 1)(x + 1)} - \frac{1(x - 1)}{(x - 1)(x + 1)}}$
Distribute the terms in the numerators:
$\frac{\frac{x^2 - x - 2}{x(x + 1)} + \frac{2x}{x(x + 1)}}{\frac{3x + 3}{(x - 1)(x + 1)} - \frac{x - 1}{(x - 1)(x + 1)}}$
Perform the operations in the numerator and in the denominator:
$\frac{\frac{x^2 - x - 2 + 2x}{x(x + 1)}}{\frac{3x + 3 - (x - 1)}{(x - 1)(x + 1)}}$
Combine like terms:
$\frac{\frac{x^2 + x - 2}{x(x + 1)}}{\frac{2x + 4}{(x - 1)(x + 1)}}$
Factor the numerators completely:
$\frac{\frac{(x + 2)(x - 1)}{x(x + 1)}}{\frac{2(x + 2)}{(x - 1)(x + 1)}}$
Rewrite this fraction as a division problem:
$\frac{(x + 2)(x - 1)}{x(x + 1)} \div \frac{2(x + 2)}{(x - 1)(x + 1)}$
To divide one fraction by another, multiply the first fraction by the reciprocal of the second fraction:
$\frac{(x + 2)(x - 1)}{x(x + 1)} \bullet \frac{(x - 1)(x + 1)}{2(x + 2)}$
Multiply the numerators together, and multiply the denominators together:
$\frac{(x + 2)(x - 1)(x - 1)(x + 1)}{2x(x + 1)(x + 2)}$
Cancel out common factors in the numerator and denominator:
$\frac{(x - 1)^2}{2x}$
