ВПР 1 №69 Алгебра = ВПТ 3 №32 Математика
Розв’яжіть рівняння:
1. $\frac{2x\ -\ 1}{3x+1}-\frac{2x\ +\ 1}{3x-5}=0;$
2. $\ 4+\frac{\ 1}{x-2}=\frac{\ 1}{2-x};$
3. $\frac{\ 8}{3x-3}+\frac{\ 2+x}{x-1}=\frac{\ 5}{2-2x}-\frac{5}{18};$
4. $\frac{2x}{x^2-1}= \frac{x}{x+1}+\frac{x}{x-1}.$
Розв'язок:
1. $\frac{\left(2x\ -\ 1\right)\left(3x-5\right)-\left(2x\ +\ 1\right)\left(3x+1\right)}{\left(3x+1\right)\left(3x-5\right)}=$
$= 0;$
$\frac{6x^2-10x-3x+5-6x^2-2x-3x-1}{\left(3x+1\right)\left(3x-5\right)}=$
$= 0;$
$\frac{-18x+4}{\left(3x+1\right)\left(3x-5\right)}=0;$
$\left\{\begin{matrix}-18x+4=0,\\x≠-\frac{1}{3},\\ x≠\frac{5}{3},\\\end{matrix}\right.$
$\left\{\begin{matrix}-18x=-4,\\x≠-\frac{1}{3},\\ x≠1\frac{2}{3},\\\end{matrix}\right.$
$x=\frac{4}{18}=\frac{2}{9}.$
2. $\frac{4\left(x-2\right)+1}{x-2}=\frac{\ 1}{2-x};$
$\frac{4x-8+1}{x-2}=\frac{\ 1}{2-x};$
$ \frac{4x-7}{x-2}-\frac{\ 1}{2-x}=0;$
$\frac{4x-7}{x-2}+\frac{\ 1}{x-2}=0;$
$ \frac{4x-7+1}{x-2}=0;$
$\frac{4x-6}{x-2}=0;$
$\left\{\begin{matrix}4x-6=0,\\x-2\neq0;\\\end{matrix}\right.$
$ \left\{\begin{matrix}4x=6,\\x\neq2;\\\end{matrix}\right.$
$\left\{\begin{matrix}x=\frac{6}{4}=\frac{3}{2}=1,5,\\x\neq2.\\\end{matrix}\right.$
3. $\frac{\ 8}{3\left(x-1\right)}+\frac{\ 2+x}{x-1}-$
$-\frac{\ 5}{2\left(1-x\right)}+\frac{5}{18}= 0;$
$\frac{\ 8}{3\left(x-1\right)}+\frac{\ 2+x}{x-1}+$
$+ \frac{\ 5}{2\left(x-1\right)}+\frac{5}{18}= 0;$
$\frac{\ 48+18\left(2+x\right)+45+5\left(x-1\right)}{18\left(x-1\right)}= 0;$
$\frac{\ 48+36+18x+45+5x-5}{18\left(x-1\right)}=0;$
$\frac{\ 23x+124}{18\left(x-1\right)}=0;$
$ \left\{\begin{matrix}23x+124=0,\\x-1\neq0;\\\end{matrix}\right.$
$ \left\{\begin{matrix}23x=-124,\\x\neq1;\\\end{matrix}\right.$
$\left\{\begin{matrix}x=-\frac{124}{23}=-5\frac{9}{23},\ \ \ \ \\x\neq1;\\\end{matrix}\right.$
$x=-5\frac{9}{23}.$
4. $\frac{2x}{\left(x-1\right)\left(x+1\right)}-$
$- \frac{\ x}{x+1}-\frac{\ x}{x-1}= 0;$
$\frac{2x-x\left(x-1\right)-x\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}= 0;$
$\frac{2x-2x^2}{\left(x-1\right)\left(x+1\right)}=0;$
$\left\{\begin{matrix}2x(1-x)=0,\\x≠1,\\ x≠-1\\\end{matrix}\right.$
$\left\{\begin{matrix}x=0 або x =1,\\x≠1,\\ x≠-1,\\\end{matrix}\right.$
$x=0.$
Відповідь:
1. $\frac{2}{9};$
2. 1,5;
3. $-5\frac{9}{23};$
4. 0.