Виконайте дію:
1) $\frac{3x(x-1)}{9x^2-y^2}+\frac{xy-y}{y^2-9x^2};$
2) $\frac{a-5}{ab-a^2}-\frac{5-b}{ab-b^2};$
3) $\frac{x^2+x-2}{x^2-4}-\frac{2}{3x-6}-1;$
4) $\frac{m^2-9}{a^2+6a+9}:\frac{12-4m}{ap+3p}.$
Розв'язок:
1) $\frac{3x\left(x-1\right)}{9x^2-y^2}+\frac{xy-y}{y^2-9x^2}=$
$= \frac{3x\left(x-1\right)}{9x^2-y^2}-\frac{y\left(x-1\right)}{9x^2-y^2}=$
$= \frac{\left(x-1\right)\left(3x-y\right)}{\left(3x-y\right)\left(3x+y\right)}=$
$=\frac{x-1}{3x+y};$
2) $\frac{a-5}{ab-a^2}-\frac{5-b}{ab-b^2}=$
$= \frac{a-5}{a\left(b-a\right)}-\frac{5-b}{b\left(a-b\right)}=$
$= \frac{a-5}{a\left(b-a\right)}+\frac{5-b}{b\left(b-a\right)}=$
$=\frac{b(a-5)+a(5-b)}{ab(b-a)}=\frac{ab-5b+5a-ab}{ab(b-a)}=$
$= \frac{5(a-b)}{-ab(a-b)}=-\frac{5}{ab};$
3) $\frac{x^2+x-2}{x^2-4}-\frac{2}{3x-6}-1=$
$= \frac{x^2+x-2}{\left(x-2\right)\left(x+2\right)}-\frac{2}{3\left(x-2\right)}-1=$
$=\frac{3\left(x^2+x-2\right)-2\left(x+2\right)-3\left(x^2-4\right)}{3\left(x-2\right)\left(x+2\right)}=$
$=\frac{3x^2+3x-6-2x-4-3x^2+12}{3(x^2-4)}=$
$= \frac{x+2}{3(x-2)(x+2)}=\frac{1}{3(x-2)};$
4) $\frac{m^2-9}{a^2+6a+9} ∶\frac{12-4m}{ap+3p}=$
$= \frac{\left(m-3\right)\left(m+3\right)}{(a+3)^2}\cdot\frac{p\left(a+3\right)}{4\left(3-m\right)}=$
$=\frac{(m-3)(m+3)\cdot p(a+3)}{-4(a+3)^2(m-3)}=-\frac{p(m+3)}{4(a+3)}.$