a

a)

\(\begin{array}{l}\frac{2}{9}:x + \frac{5}{6} = 0,5\\\frac{2}{9}:x = \frac{1}{2} - \frac{5}{6}\\\frac{2}{9}:x = \frac{3}{6} - \frac{5}{6}\\\frac{2}{9}:x = \frac{{ - 2}}{6}\\x = \frac{2}{9}:\frac{{ - 2}}{6}\\x = \frac{2}{9}.\frac{{ - 6}}{2}\\x = \frac{{ - 2}}{3}\end{array}\)                        

Vậy \(x = \frac{{ - 2}}{3}\).

b)

\(\begin{array}{l}\frac{3}{4} - \left( {x - \frac{2}{3}} \right) = 1\frac{1}{3}\\x - \frac{2}{3} = \frac{3}{4} - 1\frac{1}{3}\\x - \frac{2}{3} = \frac{3}{4} - \frac{4}{3}\\x - \frac{2}{3} = \frac{9}{{12}} - \frac{{16}}{{12}}\\x - \frac{2}{3} = \frac{{ - 7}}{{12}}\\x = \frac{{ - 7}}{{12}} + \frac{2}{3}\\x = \frac{{ - 7}}{{12}} + \frac{8}{{12}}\\x = \frac{1}{12}\end{array}\)

Vậy\(x = \frac{1}{12}\).

c)

\(\begin{array}{l}1\frac{1}{4}:\left( {x - \frac{2}{3}} \right) = 0,75\\\frac{5}{4}:\left( {x - \frac{2}{3}} \right) = \frac{3}{4}\\x - \frac{2}{3} = \frac{5}{4}:\frac{3}{4}\\x - \frac{2}{3} = \frac{5}{4}.\frac{4}{3}\\x - \frac{2}{3} = \frac{5}{3}\\x = \frac{5}{3} + \frac{2}{3}\\x = \frac{7}{3}\end{array}\)               

Vậy \(x = \frac{7}{3}\).

d)

\(\begin{array}{l}\left( { - \frac{5}{6}x + \frac{5}{4}} \right):\frac{3}{2} = \frac{4}{3}\\ - \frac{5}{6}x + \frac{5}{4} = \frac{4}{3}.\frac{3}{2}\\ - \frac{5}{6}x + \frac{5}{4} = 2\\ - \frac{5}{6}x = 2 - \frac{5}{4}\\ - \frac{5}{6}x = \frac{8}{4} - \frac{5}{4}\\ - \frac{5}{6}x = \frac{3}{4}\\x = \frac{3}{4}:\left( { - \frac{5}{6}} \right)\\x = \frac{3}{4}.\frac{{ - 6}}{5}\\x = \frac{{ - 9}}{{10}}\end{array}\)

Vậy \(x = \frac{{ - 9}}{{10}}\).

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

Đây là những bài cơ bản mà bạn!

\(\frac{5x-2}{3}=\frac{5-3x}{2}\)

\(< =>\frac{\left(5x-2\right).2}{6}=\frac{\left(5-3x\right).3}{6}\)

\(< =>\left(5x-2\right).2=\left(5-3x\right).3\)

\(< =>10x-4=15-9x\)

\(< =>10x+9x=15+4\)

\(< =>19x=19< =>x=1\)

\(\frac{10x+3}{12}=1+\frac{6+8x}{9}\)

\(< =>\frac{\left(10x+3\right).3}{36}=\frac{36}{36}+\frac{\left(6+8x\right).4}{36}\)

\(< =>\left(10x+3\right).3=36+\left(6+8x\right).4\)

\(< =>30x+9=36+24+32x\)

\(< =>32x-30x=9-36-24\)

\(< =>2x=9-60=-51< =>x=-\frac{51}{2}\)

rứt gọn ?

Câu 6 :

a, Ta có : \(x+\frac{2x+\frac{x-1}{5}}{3}=1-\frac{3x-\frac{1-2x}{3}}{5}\)

=> \(\frac{15x}{15}+\frac{5\left(2x+\frac{x-1}{5}\right)}{15}=\frac{15}{15}-\frac{3\left(3x-\frac{1-2x}{3}\right)}{15}\)

=> \(15x+5\left(2x+\frac{x-1}{5}\right)=15-3\left(3x-\frac{1-2x}{3}\right)\)

=> \(15x+10x+\frac{5\left(x-1\right)}{5}=15-9x+\frac{3\left(1-2x\right)}{3}\)

=> \(15x+10x+x-1=15-9x+1-2x\)

=> \(15x+10x+x-1-15+9x-1+2x=0\)

=> \(37x-17=0\)

=> \(x=\frac{17}{37}\)

Vậy phương trình trên có nghiệm là \(S=\left\{\frac{17}{37}\right\}\)

Bài 7 :

a, Ta có : \(\frac{x-23}{24}+\frac{x-23}{25}=\frac{x-23}{26}+\frac{x-23}{27}\)

=> \(\frac{x-23}{24}+\frac{x-23}{25}-\frac{x-23}{26}-\frac{x-23}{27}=0\)

=> \(\left(x-23\right)\left(\frac{1}{24}+\frac{1}{25}-\frac{1}{26}-\frac{1}{27}\right)=0\)

=> \(x-23=0\)

=> \(x=23\)

Vậy phương trình trên có nghiệm là \(S=\left\{23\right\}\)

c, Ta có : \(\frac{x+1}{2004}+\frac{x+2}{2003}=\frac{x+3}{2002}+\frac{x+4}{2001}\)

=> \(\frac{x+1}{2004}+1+\frac{x+2}{2003}+1=\frac{x+3}{2002}+1+\frac{x+4}{2001}+1\)

=> \(\frac{x+2005}{2004}+\frac{x+2005}{2003}=\frac{x+2005}{2002}+\frac{x+2005}{2001}\)

=> \(\frac{x+2005}{2004}+\frac{x+2005}{2003}-\frac{x+2005}{2002}-\frac{x+2005}{2001}=0\)

=> \(\left(x+2005\right)\left(\frac{1}{2004}+\frac{1}{2003}-\frac{1}{2002}-\frac{1}{2001}\right)=0\)

=> \(x+2005=0\)

=> \(x=-2005\)

Vậy phương trình trên có nghiệm là \(S=\left\{-2005\right\}\)

e, Ta có : \(\frac{x-45}{55}+\frac{x-47}{53}=\frac{x-55}{45}+\frac{x-53}{47}\)

=> \(\frac{x-45}{55}-1+\frac{x-47}{53}-1=\frac{x-55}{45}-1+\frac{x-53}{47}-1\)

=> \(\frac{x-100}{55}+\frac{x-100}{53}=\frac{x-100}{45}+\frac{x-100}{47}\)

=> \(\frac{x-100}{55}+\frac{x-100}{53}-\frac{x-100}{45}-\frac{x-100}{47}=0\)

=> \(\left(x-100\right)\left(\frac{1}{55}+\frac{1}{53}-\frac{1}{45}-\frac{1}{47}\right)=0\)

=> \(x-100=0\)

Vậy phương trình trên có nghiệm là \(S=\left\{100\right\}\)

\(\frac{4}{x^2-3x+2}-\frac{3}{2x^2-6x+1}+1=0\) \(Đkxđ:.......\)

Đặt: \(t=x^2-3x+2\left(t\ne0\right)\)

\(\Rightarrow2t=2x^2-6x+4\)

\(\Rightarrow2x^2-6x+1=2t-3\)

\(Pt:\Leftrightarrow\frac{4}{7}-\frac{3}{2t-3}+1=0\)

\(\Leftrightarrow4\left(2t-3\right)-3t+t\left(2t-3\right)=0\)

\(\Leftrightarrow8t-12-3t+2t^2-3t=0\)

\(\Leftrightarrow2t^2+2t-12=0\)

\(\Leftrightarrow\left[{}\begin{matrix}t=2\\t=-3\end{matrix}\right.\left(tm:\left[{}\begin{matrix}t\ne0\\t\ne\frac{3}{2}\end{matrix}\right.\right)\)

+ Với \(t=2\) thì: \(x^2-3x+2=2\)

\(\Leftrightarrow x\left(x-3\right)=0\)

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

+ Với \(t=-3\) thì \(x^2-3x+2=-3\)

\(\Leftrightarrow x^2-2.\frac{3}{2}x+\frac{9}{4}+\frac{11}{4}=0\)

\(\Leftrightarrow\left(x-\frac{3}{2}\right)^2+\frac{11}{4}=0\left(vô-lí\right)\)

Vậy pt có nghiệm: \(\left[{}\begin{matrix}x=0\\x=3\end{matrix}\right.\)

Bài 2:

ĐKXĐ: $x\neq 1;2;3;6$

PT $\Leftrightarrow \frac{2}{x-2}+\frac{3}{x-3}=\frac{6}{x-6}-\frac{1}{x-1}$

$\Leftrightarrow \frac{5x-12}{x^2-5x+6}=\frac{5x}{x^2-7x+6}$

Đặt $x^2+6=t$ thì $\frac{5x-12}{t-5x}=\frac{5x}{t-7x}$

$\Rightarrow (5x-12)(t-7x)=5x(t-5x)$

$\Leftrightarrow 10x^2+12t+84x=0$

$\Leftrightarrow 10x^2+12(x^2+6)+84x=0$

$\Leftrightarrow 22x^2+84x+72=0$

$\Leftrightarrow 11x^2+42x+36=0$

$\Rightarrow x=\frac{-21\pm 3\sqrt{5}}{11}$

x . \(\frac{1}{2}\)- x.\(\frac{2}{3}\) + x.\(\frac{3}{4}\)- x. \(\frac{5}{6}\) = \(\frac{5}{6}\) -\(\frac{3}{4}\) + \(\frac{2}{3}\) -\(\frac{1}{2}\)

x . \(\frac{1}{2}\)- x.\(\frac{2}{3}\) + x.\(\frac{3}{4}\)- x. \(\frac{5}{6}\) = \(\frac{10}{12}\)-\(\frac{9}{12}\)+\(\frac{8}{12}\)-\(\frac{6}{12}\)

=>x.(1/2-2/3+3/4)=1/4

=>x.7/12=1/4

=>x=1/4:7/12

=>x=1/4.12/7

=>x=3/7