Условие:
1. При каких значениях переменной имеет смысл выражение 4/(x-3)?
2. Сократите дробь:
1) (10m^8 n^3)/(15m^4 n^4 )
2) (14xy-21y)/7xy
3) (m^2-9)/(2m+6)
4) (a^2-12a+36)/(36-a^2 )
3. Выполните вычитание:
1) (y-8)/2y-(3-4y)/y^2
2) 7/a-56/(a^2+8a)
3) b/(b+1)-b^2/(b^2-1)
4) 3x-(15x^2)/(5x+2)
4. Упростите выражение:
1) (a+9)/(3a+9)-(a+3)/(3a-9)+13/(a^2-9)
2) (4b^3+8b)/(b^3-8)-2b^2/(b^2+2b+4)
5. Известно, что (a-3b)/b=4. Найдите значение выражения:
1) a/b;
2) (4a+5b)/a
6. Постройте график функции y=(4x^2-3x)/x-(x^2-4)/(x+2).
Решение:
\[\boxed{\mathbf{1}\mathbf{.}\mathbf{\ }}\]
\[\frac{4}{x - 3}\]
\[x - 3 \neq 0\]
\[x \neq 3\]
\[Выражение\ имеет\ смысл\ при\ \]
\[любом\ x,кроме\ x = 3.\]
\[\boxed{\mathbf{2}\mathbf{.}\mathbf{\ }}\]
\[1)\frac{10m^{8}n^{3}}{15m^{4}n^{4}} = \frac{2m^{4}}{3n}\]
\[2)\frac{14xy - 21y}{7xy} = \frac{7y(2x - 3)}{7xy} =\]
\[= \frac{2x - 3}{x}\]
\[3)\frac{m^{2} - 9}{2m + 6} = \frac{(m - 3)(m + 3)}{2(m + 3)} =\]
\[= \frac{m - 3}{2}\]
\[4)\frac{a^{2} - 12a + 36}{36 - a^{2}} =\]
\[= \frac{(6 - a)^{2}}{(6 - a)(6 + a)} = \frac{6 - a}{6 + a}\]
\[\boxed{\mathbf{3}\mathbf{.}\mathbf{\ }}\]
\[1)\frac{y - 8^{\backslash y}}{2y} - \frac{3 - 4y^{\backslash 2}}{y^{2}} =\]
\[= \frac{y^{2} - 8y - 6 + 8y}{2y^{2}} = \frac{y^{2} - 6}{2y^{2}}\]
\[2)\frac{7}{a} - \frac{56}{a^{2} + 8a} =\]
\[= \frac{7^{\backslash a + 8}}{a} - \frac{56}{a(a + 8)} =\]
\[= \frac{7a + 56 - 56}{a(a + 8)} = \frac{7a}{a(a + 8)} =\]
\[= \frac{7}{a + 8}\]
\[3)\frac{b^{\backslash b - 1}}{b + 1} - \frac{b^{2}}{b^{2} - 1} =\]
\[= \frac{b^{2} - b - b^{2}}{(b - 1)(b + 1)} =\]
\[= - \frac{b}{b^{2} - 1} = \frac{b}{1 - b²}\]
\[4)\ 3x^{\backslash 5x + 2} - \frac{15x^{2}}{5x + 2} =\]
\[= \frac{15x^{2} + 6x - 15x^{2}}{5x + 2} = \frac{6x}{5x + 2}\]
\[\boxed{\mathbf{4}\mathbf{.}\mathbf{\ }}\]
\[1)\frac{a + 9}{3a + 9} - \frac{a + 3}{3a - 9} + \frac{13}{a^{2} - 9} =\]
\[= \frac{3}{3(a^{2} - 9)} = \frac{1}{a^{2} - 9}\]
\[2)\frac{4b^{3} + 8b}{b^{3} - 8} - \frac{2b^{2}}{b^{2} + 2b + 4} =\]
\[= \frac{4b^{3} + 8b - 2b^{3} + 4b^{2}}{(b - 2)\left( b^{2} + 2b + 4 \right)} =\]
\[= \frac{2b^{3} + 4b^{2} + 8b}{(b - 2)\left( b^{2} + 2b + 4 \right)} =\]
\[= \frac{2b(b^{2} + 2b + 4)}{(b - 2)(b^{2} + 2b + 4)} = \frac{2b}{b - 2}\]
\[\boxed{\mathbf{5}\mathbf{.}\mathbf{\ }}\]
\[\frac{a - 3b}{b} = 4\]
\[\frac{a}{b} - \frac{3b}{b} = 4\]
\[\frac{a}{b} - 3 = 4\]
\[\frac{a}{b} = 4 + 3 = 7\]
\[1)\frac{a}{b} = 7.\]
\[2)\frac{4a + 5b}{a} = \frac{4a}{a} + \frac{5b}{a} =\]
\[= 4 + 5 \cdot \frac{b}{a} = 4 + 5 \cdot \frac{1}{7} =\]
\[= 4 + \frac{5}{7} = 4\frac{5}{7}\]
\[\boxed{\mathbf{6}\mathbf{.}\mathbf{\ }}\]
\[y = \frac{4x^{2} - 3x}{x} - \frac{x^{2} - 4}{x + 2} =\]
\[= \frac{x(4x - 3)}{x} - \frac{(x - 2)(x + 2)}{x + 2} =\]
\[= 4x - 3 - (x - 2) =\]
\[= 4x - 3 - x + 2 = 3x - 1\]
\[y = 3x - 1;\ \ \ x \neq 0;\ \ x \neq - 2\]