Упростите выражение[tex] \frac{1,5x}{x+y} + \frac{x-y}{x^2+2xy}:( \frac{y}{x^2-2xy}- \frac{2x+y}{x^2-4y^2} [/tex]. Created by ОлесяЗерина. algebra-ru

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Упростите выражение
[tex] \frac{1,5x}{x+y} + \frac{x-y}{x^2+2xy}:( \frac{y}{x^2-2xy}- \frac{2x+y}{x^2-4y^2} [/tex]

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Artem112

Verified answer

$
\frac{1,5x}{x+y} + \frac{x-y}{x^2+2xy}:( \frac{y}{x^2-2xy}- \frac{2x+y}{x^2-4y^2} )=
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=\frac{1,5x}{x+y} + \frac{x-y}{x(x+2y)}:( \frac{y}{x(x-2y)}- \frac{2x+y}{(x-2y)(x+2y)} )=
\\\
=\frac{1,5x}{x+y} + \frac{x-y}{x(x+2y)}:\frac{y(x+2y)-(2x+y)x}{x(x-2y)(x+2y)}=
\\\
=\frac{1,5x}{x+y} + \frac{x-y}{x(x+2y)}:\frac{xy+2y^2-2x^2-xy}{x(x-2y)(x+2y)}=
\\\
=\frac{1,5x}{x+y} + \frac{x-y}{x(x+2y)}:\frac{2(y-x)(y+x)}{x(x-2y)(x+2y)}=$
$=\frac{1,5x}{x+y} + \frac{x-y}{x(x+2y)}\cdot\frac{x(x-2y)(x+2y)}{2(y-x)(y+x)}=
\\\
=\frac{1,5x}{x+y} -\frac{(x-y)x(x-2y)(x+2y)}{2(x-y)(y+x)x(x+2y)}=
\\\
=\frac{3x}{2(x+y)} -\frac{x-2y}{2(y+x)}=\frac{3x-x+2y}{2(x+y)}=\frac{2x+2y}{2x+2y}=1$

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