12) \(\dfrac{2}{x+1}-\dfrac{1}{x-2}=\dfrac{3x-11}{\left(x+1\right)\left(x-2\right)}\) (ĐK: \(x\ne-1;x\ne2\))
\(\Leftrightarrow\dfrac{2\left(x-2\right)}{\left(x+1\right)\left(x-2\right)}-\dfrac{x+1}{\left(x+1\right)\left(x-2\right)}=\dfrac{3x-11}{\left(x+1\right)\left(x-2\right)}\)
\(\Leftrightarrow2x-4-x-1=3x-11\)
\(\Leftrightarrow x-5=3x-11\)
\(\Leftrightarrow3x-x=-5+11\)
\(\Leftrightarrow2x=6\)
\(\Leftrightarrow x=\dfrac{6}{2}\)
\(\Leftrightarrow x=3\left(tm\right)\)
14) \(\dfrac{x-2}{x+2}+\dfrac{3}{x-2}=\dfrac{x^2-11}{x^2-4}\) (ĐK: \(x\ne2;x\ne-2\))
\(\Leftrightarrow\dfrac{\left(x-2\right)\cdot\left(x-2\right)}{\left(x+2\right)\left(x-2\right)}+\dfrac{3\left(x+2\right)}{\left(x+2\right)\left(x-2\right)}=\dfrac{x^2-11}{\left(x+2\right)\left(x-2\right)}\)
\(\Leftrightarrow x^2-4x+4+3x+6=x^2-11\)
\(\Leftrightarrow x^2-x+10=x^2-11\)
\(\Leftrightarrow x^2-x^2-x=-11+10\)
\(\Leftrightarrow-x=-21\)
\(\Leftrightarrow x=21\left(tm\right)\)
16) \(\dfrac{x-3}{x-2}+\dfrac{x+2}{x}=2\) (ĐK: \(x\ne2,x\ne0\))
\(\Leftrightarrow\dfrac{x\left(x-3\right)}{x\left(x-2\right)}+\dfrac{\left(x+2\right)\left(x-2\right)}{x\left(x-2\right)}=\dfrac{2x\left(x-2\right)}{x\left(x-2\right)}\)
\(\Leftrightarrow x^2-3x+x^2-4=2x^2-4x\)
\(\Leftrightarrow2x^2-3x-4=2x^2-4x\)
\(\Leftrightarrow2x^2-2x^2-3x+4x=4\)
\(\Leftrightarrow x=4\left(tm\right)\)
18) \(\dfrac{x+4}{x+1}+\dfrac{x}{x-1}=\dfrac{2x^2}{x^2-1}\) (ĐK: \(x\ne1,x\ne-1\))
\(\Leftrightarrow\dfrac{\left(x+4\right)\left(x-1\right)}{\left(x+1\right)\left(x-1\right)}+\dfrac{x\left(x+1\right)}{\left(x+1\right)\left(x-1\right)}=\dfrac{2x^2}{\left(x+1\right)\left(x-1\right)}\)
\(\Leftrightarrow x^2-x+4x-4+x^2+x=2x^2\)
\(\Leftrightarrow2x^2+4x-4=2x^2\)
\(\Leftrightarrow2x^2-2x^2+4x=4\)
\(\Leftrightarrow x=\dfrac{4}{4}\)
\(\Leftrightarrow x=1\left(ktm\right)\)
Vậy pt vô nghiệm
20) \(\dfrac{2x-3}{x-1}+1=\dfrac{6x-x^2-6}{x-1}\) (ĐK: \(x\ne1\))
\(\Leftrightarrow\dfrac{6x-x^2-6}{x-1}-\dfrac{2x-3}{x-1}=1\)
\(\Leftrightarrow\dfrac{6x-x^2-6-2x+3}{x-1}=1\)
\(\Leftrightarrow4x-x^2-3=x-1\)
\(\Leftrightarrow x^2-3x+2=0\)
\(\Leftrightarrow x^2-x-2x+2=0\)
\(\Leftrightarrow\left(x-1\right)\left(x-2\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=1\left(ktm\right)\\x=2\left(tm\right)\end{matrix}\right.\)
\(\Leftrightarrow x=2\)
18:
ĐKXĐ: x<>1; x<>-1
\(\dfrac{x+4}{x+1}+\dfrac{x}{x-1}=\dfrac{2x^2}{x^2-1}\)
=>(x+4)(x-1)+x(x+1)=2x^2
=>x^2+3x-4+x^2+x=2x^2
=>4x-4=0
=>x=1(loại)
20: ĐKXĐ: x<>1
\(\dfrac{2x-3}{x-1}+1=\dfrac{6x-x^2-6}{x-1}\)
=>2x-3+x-1=6x-x^2-6
=>3x-4=-x^2+6x-6
=>x^2+3x-4-6x+6=0
=>x^2-3x+2=0
=>(x-1)(x-2)=0
=>x=1(loại) hoặc x=2(nhận)
22: ĐKXĐ: x<>1; x<>-1
\(\dfrac{1}{2x-2}-\dfrac{2x-1}{x^2+x+1}+\dfrac{3}{2x+2}=0\)
=>\(\dfrac{x+1+3\left(x-1\right)}{2\left(x-1\right)\left(x+1\right)}-\dfrac{2x-1}{x^2+x+1}=0\)
=>\(\dfrac{4x-2}{2\left(x-1\right)\left(x+1\right)}-\dfrac{2x-1}{x^2+x+1}=0\)
=>\(\dfrac{2x-1}{\left(x-1\right)\left(x+1\right)}-\dfrac{2x-1}{x^2+x+1}=0\)
=>\(\left(2x-1\right)\left(\dfrac{1}{x^2-1}-\dfrac{1}{x^2+x+1}\right)=0\)
=>\(\left(2x-1\right)\cdot\dfrac{x^2+x+1-x^2+1}{\left(x-1\right)\left(x+1\right)\left(x^2+x+1\right)}=0\)
=>(2x-1)(x+2)=0
=>x=1/2 hoặc x=-2
24: ĐKXĐ: x<>0
\(\dfrac{x^2-4}{x}=\dfrac{2x+3}{2}\)
=>2(x^2-4)=x(2x+3)
=>2x^2-8=2x^2+3x
=>x=-8/3(nhận)
