Ответы к странице 27

102. Выполните действия:
1) $\frac{2}{x} + \frac{3x - 2}{x + 1}$;
2) $\frac{m}{n} - \frac{m}{m + n}$;
3) $\frac{a}{a - 3} - \frac{3}{a + 3}$;
4) $\frac{c}{3c - 1} - \frac{c}{3c + 1}$;
5) $\frac{x}{2y + 1} - \frac{x}{3y - 2}$;
6) $\frac{a - b}{b} - \frac{a - b}{a + b}$.

Решение:

1) $\frac{2}{x} + \frac{3x - 2}{x + 1} = \frac{2(x + 1) + x(3x - 2)}{x(x + 1)} = \frac{2x + 2 + 3x^2 - 2x}{x(x + 1)} = \frac{3x^2 + 2}{x(x + 1)}$

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

3) $\frac{a}{a - 3} - \frac{3}{a + 3} = \frac{a(a + 3) - 3(a - 3)}{(a - 3)(a + 3)} = \frac{a^2 + 3a - 3a + 9}{(a - 3)(a + 3)} = \frac{a^2 + 9}{a^2 - 9}$

4) $\frac{c}{3c - 1} - \frac{c}{3c + 1} = \frac{c(3c + 1) - c(3c - 1)}{(3c - 1)(3c + 1)} = \frac{3c^2 + c - 3c^2 + c}{(3c - 1)(3c + 1)} = \frac{2c}{9c^2 - 1}$

5) $\frac{x}{2y + 1} - \frac{x}{3y - 2} = \frac{x(3y - 2) - x(2y + 1)}{(2y + 1)(3y - 2)} = \frac{3xy - 2x - 2xy - x}{(2y + 1)(3y - 2)} = \frac{xy - 3x}{(2y + 1)(3y - 2)}$

6) $\frac{a - b}{b} - \frac{a - b}{a + b} = \frac{(a - b)(a + b) - b(a - b)}{b(a + b)} = \frac{a^2 - b^2 - ab + b^2}{b(a + b)} = \frac{a^2 - ab}{b(a + b)}$

103. Представьте в виде дроби выражение:
1) $\frac{a}{a - b} + \frac{a}{b}$;
2) $\frac{4}{x} - \frac{5x + 4}{x + 2}$;
3) $\frac{b}{b - 2} - \frac{2}{b + 2}$.

Решение:

1) $\frac{a}{a - b} + \frac{a}{b} = \frac{ab + a(a - b)}{b(a - b)} = \frac{ab + a^2 - ab}{b(a - b)} = \frac{a^2}{b(a - b)}$

2) $\frac{4}{x} - \frac{5x + 4}{x + 2} = \frac{4(x + 2) - x(5x + 4)}{x(x + 2)} = \frac{4x + 8 - 5x^2 - 4x}{x(x + 2)} = \frac{8 - 5x^2}{x(x + 2)}$

3) $\frac{b}{b - 2} - \frac{2}{b + 2} = \frac{b(b + 2) - 2(b - 2)}{(b - 2)(b + 2)} = \frac{b^2 + 2b - 2b + 4}{(b - 2)(b + 2)} = \frac{b^2 + 4}{b^2 - 4}$

104. Упростите выражение:
1) $\frac{1}{b(a - b)} - \frac{1}{a(a - b)}$;
2) $\frac{5}{a} + \frac{30}{a(a - 6)}$;
3) $\frac{3}{x - 2} - \frac{2x + 2}{x(x - 2)}$;
4) $\frac{y}{2(y + 3)} - \frac{y}{5(y + 3)}$;
5) $\frac{5m + 3}{2(m + 1)} - \frac{7m + 4}{3(m + 1)}$;
6) $\frac{c - a}{a(a + b)} + \frac{c + b}{b(a + b)}$.

Решение:

1) $\frac{1}{b(a - b)} - \frac{1}{a(a - b)} = \frac{a - b}{ab(a - b)} = \frac{1}{ab}$

2) $\frac{5}{a} + \frac{30}{a(a - 6)} = \frac{5(a - 6) + 30}{a(a - 6)} = \frac{5a - 30 + 30}{a(a - 6)} = \frac{5a}{a(a - 6)} = \frac{5}{a - 6}$

3) $\frac{3}{x - 2} - \frac{2x + 2}{x(x - 2)} = \frac{3x - (2x + 2)}{x(x - 2)} = \frac{3x - 2x - 2}{x(x - 2)} = \frac{x - 2}{x(x - 2)} = \frac{1}{x}$

4) $\frac{y}{2(y + 3)} - \frac{y}{5(y + 3)} = \frac{5y - 2y}{10(y + 3)} = \frac{3y}{10(y + 3)}$

5) $\frac{5m + 3}{2(m + 1)} - \frac{7m + 4}{3(m + 1)} = \frac{3(5m + 3) - 2(7m + 4)}{6(m + 1)} = \frac{15m + 9 - 14m - 8}{6(m + 1)} = \frac{m + 1}{6(m + 1)} = \frac{1}{6}$

6) $\frac{c - a}{a(a + b)} + \frac{c + b}{b(a + b)} = \frac{b(c - a) + a(c + b)}{ab(a + b)} = \frac{bc - ab + ac + ab}{ab(a + b)} = \frac{bc + ac}{ab(a + b)} = \frac{c(a + b)}{ab(a + b)} = \frac{c}{ab}$

105. Выполните действия:
1) $\frac{1}{a(a + b)} + \frac{1}{b(a + b)}$;
2) $\frac{4}{b} - \frac{8}{b(b + 2)}$;
3) $\frac{x}{5(x + 7)} - \frac{x}{6(x + 7)}$;
4) $\frac{4n + 2}{3(n - 1)} - \frac{5n + 3}{4(n - 1)}$.

Решение:

1) $\frac{1}{a(a + b)} + \frac{1}{b(a + b)} = \frac{b + a}{ab(a + b)} = \frac{1}{ab}$

2) $\frac{4}{b} - \frac{8}{b(b + 2)} = \frac{4(b + 2) - 8}{b(b + 2)} = \frac{4b + 8 - 8}{b(b + 2)} = \frac{4b}{b(b + 2)} = \frac{4}{b + 2}$

3) $\frac{x}{5(x + 7)} - \frac{x}{6(x + 7)} = \frac{6x - 5x}{30(x + 7)} = \frac{x}{30(x + 7)}$

4) $\frac{4n + 2}{3(n - 1)} - \frac{5n + 3}{4(n - 1)} = \frac{4(4n + 2) - 3(5n + 3)}{12(n - 1)} = \frac{16n + 8 - 15n - 9}{12(n - 1)} = \frac{n - 1}{12(n - 1)} = \frac{1}{12}$

106. Выполните сложение или вычитание дробей:
1) $\frac{a}{a - 2} - \frac{3a + 1}{3a - 6}$;
2) $\frac{18}{b^2 + 3b} - \frac{6}{b}$;
3) $\frac{2}{c + 1} - \frac{c - 1}{c^2 + c}$;
4) $\frac{d - 1}{2d - 8} + \frac{d}{d - 4}$;
5) $\frac{m + 1}{3m - 15} - \frac{m - 1}{2m - 10}$;
6) $\frac{m - 2n}{6m + 6n} - \frac{m - 3n}{4m + 4n}$;
7) $\frac{a^2 + 2}{a^2 + 2a} - \frac{a + 4}{2a + 4}$;
8) $\frac{3x - 4y}{x^2 - 2xy} - \frac{3y - x}{xy - 2y^2}$.

Решение:

1) $\frac{a}{a - 2} - \frac{3a + 1}{3a - 6} = \frac{a}{a - 2} - \frac{3a + 1}{3(a - 2)} = \frac{3a - (3a + 1)}{3(a - 2)} = \frac{3a - 3a - 1}{3(a - 2)} = \frac{-1}{3(a - 2)} = -\frac{1}{3(a - 2)}$

2) $\frac{18}{b^2 + 3b} - \frac{6}{b} = \frac{18}{b(b + 3)} - \frac{6}{b} = \frac{18 - 6(b + 3)}{b(b + 3)} = \frac{18 - 6b - 18}{b(b + 3)} = \frac{-6b}{b(b + 3)} = -\frac{6}{b + 3}$

3) $\frac{2}{c + 1} - \frac{c - 1}{c^2 + c} = \frac{2}{c + 1} - \frac{c - 1}{c(c + 1)} = \frac{2c - (c - 1)}{c(c + 1)} = \frac{2c - (c - 1)}{c(c + 1)} = \frac{2c - c + 1}{c(c + 1)} = \frac{c + 1}{c(c + 1)} = \frac{1}{c}$

4) $\frac{d - 1}{2d - 8} + \frac{d}{d - 4} = \frac{d - 1}{2(d - 4)} + \frac{d}{d - 4} = \frac{d - 1 + 2d}{2(d - 4)} = \frac{3d - 1}{2(d - 4)}$

5) $\frac{m + 1}{3m - 15} - \frac{m - 1}{2m - 10} = \frac{m + 1}{3(m - 5)} - \frac{m - 1}{2(m - 5)} = \frac{2(m + 1) - 3(m - 1)}{6(m - 5)} = \frac{2m + 2 - 3m + 3}{6(m - 5)} = \frac{5 - m}{6(m - 5)} = -\frac{m - 5}{6(m - 5)} = -\frac{1}{6}$

6) $\frac{m - 2n}{6m + 6n} - \frac{m - 3n}{4m + 4n} = \frac{m - 2n}{6(m + n)} - \frac{m - 3n}{4(m + n)} = \frac{2(m - 2n) - 3(m - 3n)}{12(m + n)} = \frac{2m - 4n - 3m + 9n}{12(m + n)} = \frac{5n - m}{12(m + n)}$

7) $\frac{a^2 + 2}{a^2 + 2a} - \frac{a + 4}{2a + 4} = \frac{a^2 + 2}{a(a + 2)} - \frac{a + 4}{2(a + 2)} = \frac{2(a^2 + 2) - a(a + 4)}{2a(a + 2)} = \frac{2a^2 + 4 - a^2 - 4a}{2a(a + 2)} = \frac{a^2 - 4a + 4}{2a(a + 2)} = \frac{(a - 2)^2}{2(a + 2)}$

8) $\frac{3x - 4y}{x^2 - 2xy} - \frac{3y - x}{xy - 2y^2} = \frac{3x - 4y}{x(x - 2y)} - \frac{3y - x}{y(x - 2y)} = \frac{y(3x - 4y) - x(3y - x)}{xy(x - 2y)} = \frac{3xy - 4y^2 - 3xy + x^2}{xy(x - 2y)} = \frac{x^2 - 4y^2}{xy(x - 2y)} = \frac{(x - 2y)(x + 2y)}{xy(x - 2y)} = \frac{x + 2y}{xy}$

107. Упростите выражение:
1) $\frac{b}{b - 5} - \frac{4b - 1}{4b - 20}$;
2) $\frac{2}{m} - \frac{16}{m^2 + 8m}$;
3) $\frac{a - 2}{2a - 6} - \frac{a - 1}{3a - 9}$;
4) $\frac{a^2 + b^2}{2a^2 + 2ab} + \frac{b}{a + b}$;
5) $\frac{b + 4}{ab - b^2} - \frac{a + 4}{a^2 - ab}$;
6) $\frac{c - 4}{4c + 24} + \frac{4c + 9}{c^2 + 6c}$.

Решение:

1) $\frac{b}{b - 5} - \frac{4b - 1}{4b - 20} = \frac{b}{b - 5} - \frac{4b - 1}{4(b - 5)} = \frac{4b - (4b - 1)}{4(b - 5)} = \frac{4b - 4b + 1}{4(b - 5)} = \frac{1}{4(b - 5)}$

2) $\frac{2}{m} - \frac{16}{m^2 + 8m} = \frac{2}{m} - \frac{16}{m(m + 8)} = \frac{2(m + 8) - 16}{m(m + 8)} = \frac{2m + 16 - 16}{m(m + 8)} = \frac{2m}{m(m + 8)} = \frac{2}{m + 8}$

3) $\frac{a - 2}{2a - 6} - \frac{a - 1}{3a - 9} = \frac{a - 2}{2(a - 3)} - \frac{a - 1}{3(a - 3)} = \frac{3(a - 2) - 2(a - 1)}{6(a - 3)} = \frac{3a - 6 - 2a + 2}{6(a - 3)} = \frac{a - 4}{6(a - 3)}$

4) $\frac{a^2 + b^2}{2a^2 + 2ab} + \frac{b}{a + b} = \frac{a^2 + b^2}{2a(a + b)} + \frac{b}{a + b} = \frac{a^2 + b^2 + 2ab}{2a(a + b)} = \frac{(a + b)^2}{2a(a + b)} = \frac{a + b}{2a}$

5) $\frac{b + 4}{ab - b^2} - \frac{a + 4}{a^2 - ab} = \frac{b + 4}{b(a - b)} - \frac{a + 4}{a(a - b)} = \frac{a(b + 4) - b(a + 4)}{ab(a - b)} = \frac{ab + 4a - ab - 4b)}{ab(a - b)} = \frac{4a - 4b}{ab(a - b)} = \frac{4(a - b)}{ab(a - b)} = \frac{4}{ab}$

6) $\frac{c - 4}{4c + 24} + \frac{4c + 9}{c^2 + 6c} = \frac{c - 4}{4(c + 6)} + \frac{4c + 9}{c(c + 6)} = \frac{c(c - 4) + 4(4c + 9)}{4c(c + 6)} = \frac{c^2 - 4c + 16c + 36}{4c(c + 6)} = \frac{c^2 + 12c + 36}{4c(c + 6)} = \frac{(c + 6)^2}{4c(c + 6)} = \frac{c + 6}{4c}$