\(\dfrac{x-1}{x}-\dfrac{1}{x+1}=\dfrac{2x-1}{x^2+x}\)

\(\Leftrightarrow\dfrac{x-1}{x}-\dfrac{1}{x+1}=\dfrac{2x-1}{x\left(x+1\right)}\)

ĐKXĐ : \(\left\{{}\begin{matrix}x\ne0\\x+1\ne0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x\ne0\\x\ne-1\end{matrix}\right.\)

Ta có : `(x-1)/x -1/(x+1) =(2x-1)/(x(x+1))`

\(\Leftrightarrow\dfrac{\left(x-1\right)\left(x+1\right)}{x\left(x+1\right)}-\dfrac{x}{x\left(x+1\right)}=\dfrac{2x-1}{x\left(x+1\right)}\)

`=> x^2 +x -x-1 -x-2x+1=0`

`<=> x^2 -3x =0`

`<=> x(x-3)=0`

\(\Leftrightarrow\left[{}\begin{matrix}x=0\\x-3=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=0\left(ktm\right)\\x=3\end{matrix}\right.\)

__

`(x+2)(5-3x)=0`

\(\Leftrightarrow\left[{}\begin{matrix}x+2=0\\5-3x=0\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=-2\\3x=5\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=-2\\x=\dfrac{5}{3}\end{matrix}\right.\)

__

\(\dfrac{5\left(1-2x\right)}{3}+\dfrac{x}{2}=\dfrac{3\left(x-5\right)}{4}-2\)

\(\Leftrightarrow\dfrac{20\left(1-2x\right)}{12}+\dfrac{6x}{12}=\dfrac{9\left(x-5\right)}{12}-\dfrac{24}{12}\)

`<=> 2x- 40x + 6x = 9x - 45 -24`

`<=>  2x- 40x + 6x-9x + 45 +24=0`

`<=>-41x+69=0`

`<=>-41x=-69`

`<=> x=69/41`

Cậu tách 2 câu 1 lượt mn trl nhanh hơn đó ạ

a:=>x^2-1-x=2x-1

=>x^2-x-1=2x-1

=>x^2-3x=0

=>x=0(loại) hoặc x=3(nhận)

b:=>x+2=0 hoặc 5-3x=0

=>x=-2 hoặc x=5/3

c:=>20(1-2x)+6x=9(x-5)-24

=>20-40x+6x=9x-45-24

=>-34x+20=9x-69

=>-43x=-89

=>x=89/43

d: =>x^2+4x+4-x^2-2x+3=2x^2+8x-4x-16-3

=>2x^2+4x-19=-2x+7

=>2x^2+6x-26=0

=>x^2+3x-13=0

=>\(x=\dfrac{-3\pm\sqrt{61}}{2}\)

e: =>(2x-3)(2x-3-x-1)=0

=>(2x-3)(x-4)=0

=>x=4 hoặc x=3/2

a) \(\frac{\left(5.2\right)}{3.2}-\frac{1}{2}x+\frac{1}{3}+\frac{1}{5}=\frac{\left(3.2\right)}{5}\)

\(\Leftrightarrow\)\(\frac{1}{2}-\frac{1}{2}x+\frac{8}{15}=\frac{6}{5}\)

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

\(\Leftrightarrow\)\(-\frac{1}{6}=\frac{1}{2}x\)

\(\Leftrightarrow\)x=-1/3

b) VT= \(\frac{\left(3.5.4.2\right)}{5.2.3}=4\)

\(\Leftrightarrow\left(x-\frac{1}{2}\right):6+4=4:\frac{2}{3}=6\)

\(\Leftrightarrow\left(x-\frac{1}{2}\right):6=2\)

\(\Leftrightarrow x-\frac{1}{2}=12\)

=> x= 12,5

thiếu dữ liệu đề bài bạn ơi

bài này k có số kết thúc thì k giải dc

Đây là dãy lùi vô hạn và có công thức tình đàng hoàn nhé:

\(A=\frac{7}{10}+\frac{7}{10^2}+\frac{7}{10^3}+...\)

\(=7.\left(\frac{1}{10}+\frac{1}{10^2}+\frac{1}{10^3}+...\right)=7.\frac{\frac{1}{10}}{1-\frac{1}{10}}=\frac{7}{9}\)

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

=\(\frac{2}{3}+\frac{1}{2}+\frac{1}{4}\)

=\(\frac{17}{12}\)

Đề gõ sai, xin sửa lại:
Chứng minh:

\({1 \over {11}^2} + {1 \over {12}^2} + {1 \over {13}^2} + {1 \over {14}^2} + ... + {1 \over {100}^2}<{1 \over {10}}\)

Cảm ơn

Đặt biểu thức là A     ta có:

1/11^2 < 1/10.11 = 1/10 - 1/11

1/12^2 < 1/11.12 = 1/11 - 1/12

1/13^2 < 1/12.13 = 1/12 - 1/13

. . . . . . . . . 

1/100^2 < 1/99.100 = 1/99 - 1/100

 => A < 1/10 - 1/11 + 1/11 - 1/12 + 1/12 - 1/13 + . . . .+ 1/99 - 1/100

  => A < 1/10 -  1/100

 => A < 1/10

                Bạn nhớ k cho mình nha

Có\(\frac{1}{11^2}\)<\(\frac{1}{10.11}\);...;\(\frac{1}{100}\)<\(\frac{1}{99.100}\)\(\Rightarrow\)\(\frac{1}{11^2}\)+...+\(\frac{1}{100^2}\)<\(\frac{1}{10}\)-\(\frac{1}{11}\)+\(\frac{1}{11}\)-\(\frac{1}{12}\)+.......+\(\frac{1}{99}\)-\(\frac{1}{100}\)<\(\frac{1}{10}\)-\(\frac{1}{100}\)

Ta có  4A=\(1+\frac{1}{2^2}+\frac{1}{2^4}+...+\frac{1}{2^{98}}\)

Trừ 4A cho A ta được 

3A = \(1-\frac{1}{2^{100}}\)=> 3A <1 => A<1/3 (đpcm)

Chúc bạn học tốt 

Ta có :\(A=\frac{1}{2^2}+...+\frac{1}{2^{100}}\)

\(2A=\frac{1}{2}+...+\frac{1}{2^{99}}\)

\(2A-A=\left(\frac{1}{2}+...+\frac{1}{2^{99}}\right)-\left(\frac{1}{2^2}+...+\frac{1}{2^{100}}\right)\)

\(A=\frac{1}{2}-\frac{1}{2^{100}}\)

Lại có :

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

Vì \(\frac{1}{2^{100}}< \frac{1}{6}\)

\(\Rightarrow\frac{1}{2}-\frac{1}{2^{100}}>\frac{1}{2}-\frac{1}{6}\)

\(\Rightarrow A>\frac{1}{3}\)

Vậy \(A>\frac{1}{3}\)(ĐPCM)