Simplify the following expression: (4x)/(y2 - x2) + 2/(x+4) − 3/(y-x)

Question

ScanSolve AI Tutor · Accepted Answer

Here's the solution for the expression:

\frac{4x}{y^2 - x^2} + \frac{2}{x+y} - \frac{3}{y-x}

Factor the denominators.
The first denominator $y^2 - x^2$ is a difference of squares, which factors to $(y-x)(y+x)$ .
The second denominator is $x+y$ , which is the same as $y+x$ .
The third denominator is $y-x$ .
Find the least common denominator (LCD).
The factors are $(y-x)$ and $(y+x)$ . Therefore, the LCD is $(y-x)(y+x)$ .
Rewrite each fraction with the LCD.
$\frac{4x}{(y-x)(y+x)} + \frac{2(y-x)}{(x+y)(y-x)} - \frac{3(y+x)}{(y-x)(y+x)}$
Combine the numerators over the common denominator.
$\frac{4x + 2(y-x) - 3(y+x)}{(y-x)(y+x)}$
Expand and simplify the numerator.
$\frac{4x + 2y - 2x - 3y - 3x}{(y-x)(y+x)}$
Combine the $x$ terms and $y$ terms:
$\frac{(4x - 2x - 3x) + (2y - 3y)}{(y-x)(y+x)}$ $\frac{-x - y}{(y-x)(y+x)}$
Factor out $-1$ from the numerator and simplify.
$\frac{-(x+y)}{(y-x)(y+x)}$
Since $(x+y)$ is the same as $(y+x)$ , we can cancel this term from the numerator and denominator.
$\frac{-1}{y-x}$
This can also be written as:
$\frac{1}{-(y-x)} = \frac{1}{x-y}$
The simplified expression is:
$\frac{1}{x-y}$

Simplify the following expression: (4x)/(y2 - x2) + 2/(x+4) - 3/(y-x)