Упражнение 86

\({\largeа)}\ \frac{b-c}{b}+\frac{b}{b+c}=\frac{ (b-c)(b+c)+b\cdot{b}}{ b(b+c)}=\frac{b^2-c^2+b^2}{ b(b+c)}=\frac{2b^2-c^2}{b^2+bc};\)

\({\largeб)}\ \frac{x+1}{x-2}-\frac{x+3}{x}=\frac{ (x+1)\cdot{x}-(x+3)(x-2)}{ x(x-2)}=\frac{x^2+x-(x^2-2x+3x-6)}{ x(x-2)}=\frac{x^2+x-x^2+2x-3x+6}{ x(x-2)}=\frac{6}{x^2-2x};\)

\({\largeв)}\ \frac{m}{m-n}-\frac{n}{m+n}=\frac{ m(m+n)-n(m-n)}{ (m-n)(m+n)}=\frac{m^2+mn-mn+n^2}{ (m-n)(m+n)}=\frac{m^2+n^2}{m^2-n^2};\)

\({\largeг)}\ \frac{2a}{2a-1}-\frac{1}{2a+1}=\frac{ 2a(2a+1)-(2a-1)}{ (2a-1)(2a+1)}=\frac{4a^2+2a-2a+1}{ (2a-1)(2a+1)}=\frac{4a^2+1}{4a^2-1};\)

\({\largeд)}\ \frac{a}{a+2}-\frac{a}{a-2}=\frac{ a(a-2)-a(a+2)}{ (a-2)(a+2)}=\frac{a^2-2a-a^2-2a}{ (a-2)(a+2)}=\frac{-4a}{ (a-2)(a+2)}={-}\frac{4a}{a^2-4};\)

\({\largeе)}\ \frac{p}{3p-1}-\frac{p}{1+3p}=\frac{ p(3p+1)-p(3p-1)}{ (3p-1)(3p+1)}=\frac{3p^2+p-3p^2+p}{ (3p-1)(3p+1)}=\frac{2p}{9p^2-1}.\)