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\(\frac{2}{x^3-x^2-x+1}=\frac{3}{1-x^2}-\frac{1}{x+1}\)
<=> \(\frac{2}{\left(x^2-1\right)\left(x-1\right)}+\frac{3}{\left(x-1\right)\left(x+1\right)}+\frac{1}{x+1}=0\)
<=> \(\frac{2}{\left(x-1\right)^2\left(x+1\right)}+\frac{3\left(x-1\right)}{\left(x-1\right)^2\left(x+1\right)}+\frac{\left(x-1\right)^2}{\left(x-1\right)^2\left(x+1\right)}=0\)
<=> \(2+3x-3+x^2-2x+1=0\)
<=> x2 + x = 0
<=> x(x + 1) = 0
<=> \(\orbr{\begin{cases}x=0\\x+1=0\end{cases}}\)
<=> \(\orbr{\begin{cases}x=0\\x=-1\end{cases}}\)
Vậy S = {0; -1}
\(\frac{x+2}{x+1}+\frac{3}{x-2}=\frac{3}{x^2-x-2}+1\)
\(\frac{\left(x+2\right)\left(x-2\right)+3\left(x+1\right)}{x^2-x-2}=\frac{3+x^2-x-2}{x^2-x-2}\)
\(x^2-4+3x+3=1+x^2-x\)
\(x^2+3x-1-1-x^2+x=0\)
\(4x-2=0\)
\(4x=2\Leftrightarrow x=\frac{1}{2}\)
Vậy.....
\(\frac{x+2}{x+1}+\frac{3}{x-2}=\frac{3}{x^2-x-2}+1\)
\(\Leftrightarrow\)\(\frac{x+2}{x+1}+\frac{3}{x-2}=\frac{3}{\left(x+1\right).\left(x-2\right)}+1\)
ĐKXĐ: \(x\ne-1,2\)
\(\frac{\left(x+2\right).\left(x-2\right)}{\left(x+1\right).\left(x-2\right)}+\)\(\frac{3.\left(x+1\right)}{\left(x+1\right).\left(x-2\right)}=\)\(\frac{3}{\left(x+1\right).\left(x-2\right)}+\frac{\left(x+1\right).\left(x-2\right)}{\left(x+1\right).\left(x-2\right)}\)
\(\Leftrightarrow\) \(\left(x^2-4\right)\) \(+3.\left(x+1\right)=\)\(3+\left(x+1\right).\left(x-2\right)\)
\(\Leftrightarrow\) x2 - 4 + 3x + 3 = 3 + x2 - x - 2
\(\Leftrightarrow\) x2 + 3x - x2 + x = 4 - 3 + 3 - 2
\(\Leftrightarrow\) 4x = 2
\(\Leftrightarrow\)\(x=\frac{1}{2}\)
Vậy phương trình có nghiệm là: \(x=\frac{1}{2}\)
\(x-\frac{\frac{x}{2}-\frac{3+x}{4}}{2}=3-\frac{\left(1-\frac{6-x}{3}\right).\frac{1}{2}}{2}\)
\(\Leftrightarrow2x-\frac{x}{2}+\frac{3+x}{4}=6-\frac{1}{2}+\frac{6-x}{6}\)
\(\Leftrightarrow24x-6x+9+3x=72-6+12-2x\)
\(\Leftrightarrow23x=69\)
\(\Leftrightarrow x=3\)
Vậy nghiệm của pt x=3
a) \(\frac{x-1}{2}+\frac{x-2}{3}+\frac{x-3}{4}=\frac{x-4}{5}+\frac{x-5}{6}\)
\(\left(\frac{x-1}{2}+1\right)+\left(\frac{x-2}{3}+3\right)+\left(\frac{x-3}{4}+1\right)=\left(\frac{x-4}{5}+1\right)+\left(\frac{x-5}{6}+1\right)\)
\(\frac{x-1}{2}+\frac{x-1}{3}+\frac{x-1}{4}=\frac{x-1}{5}+\frac{x-1}{6}\)
\(\left(x-1\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}\right)\)=0
\(x-1=0\)
\(x=1\)
\(a,\Leftrightarrow5\left(x-2\right)-15x\le9+10\left(x+1\right)\)
\(\Leftrightarrow5x-10-15x\le9+10x+10\)
\(\Leftrightarrow-20x\le29\)
\(\Leftrightarrow x\ge-1,45\)
Vậy ...........
\(b,\Rightarrow\left(x+2\right)-3\left(x-3\right)=5\left(x-2\right)\)
\(\Leftrightarrow x+2-3x+9-5x+10=0\)
\(\Leftrightarrow-7x+21=0\)
\(\Leftrightarrow x=3\)
Vậy ..............
\(\frac{x-2}{6}-\frac{x}{2}\le\frac{3}{10}+\frac{x+1}{3}\Leftrightarrow\frac{5\left(x-2\right)}{30}-\frac{15x}{30}\le\frac{9}{30}+\frac{10\left(x+1\right)}{30}\)
\(\Leftrightarrow5x-10-15x-9-10x-10\le0\)
\(\Leftrightarrow-20x-29\le0\Leftrightarrow\left(-20x\right)\cdot\frac{-1}{20}\ge29\cdot-\frac{1}{20}\)
\(\Leftrightarrow x\ge-\frac{29}{20}\)
\(a,\frac{3}{x^2+x-2}-\frac{1}{x-1}=\frac{-7}{x+2}\left(x\ne1;x\ne-2\right)\)
\(\Leftrightarrow\frac{3}{x^2+x-2}-\frac{1}{x-1}+\frac{7}{x+2}=0\)
\(\Leftrightarrow\frac{3}{\left(x-1\right)\left(x+2\right)}-\frac{1\left(x+2\right)}{\left(x-1\right)\left(x+2\right)}+\frac{7\left(x-1\right)}{\left(x-1\right)\left(x+2\right)}=0\)
\(\Leftrightarrow\frac{3}{\left(x-1\right)\left(x+2\right)}-\frac{x+2}{\left(x-1\right)\left(x+2\right)}+\frac{7x-7}{\left(x-1\right)\left(x+2\right)}=0\)
\(\Leftrightarrow\frac{3-x-2+7x-7}{\left(x-1\right)\left(x+2\right)}=0\)
\(\Leftrightarrow\frac{6x-8}{\left(x-1\right)\left(x+2\right)}=0\)
=> 6x-8=0
<=> x=\(\frac{8}{6}=\frac{4}{3}\left(tmđk\right)\)
b) ĐKXĐ: x khác 2; x khác 4
\(\frac{2}{-x^2+6x-8}-\frac{x-1}{x-2}=\frac{x+3}{x-4}\)
<=> \(\frac{2}{\left(x-2\right)\left(x-4\right)}+\frac{x-1}{x-2}=\frac{x+3}{x-4}\)
<=> 2(x - 2) + (x - 1)(x - 4)(x - 2) = (x + 3)(x - 2)(x - 2)
<=> x^3 - 7x^2 + 16x - 12 = -x^3 + x^2 + 8x - 12
<=> x^2 - 7x^2 + 16x - 12 + x^3 - x^2 + 8x - 12 = 0
<=> 2x^3 - 8x^2 + 8x = 0
<=> 2x(x - 2)(x - 2) = 0
<=> 2x = 0 hoặc x - 2 = 0
<=> x = 0 (tmđk) hoặc x = 2 (ktmđk)
=> x = 2
ĐKXĐ \(x\ne0,-1,-2,...,-100\)
\(\frac{1}{x^2+x}+\frac{1}{x^2+3x+2}+...+\frac{1}{x^2+199x+9900}=\frac{25}{51}\)
\(\Leftrightarrow\frac{1}{x\left(x+1\right)}+\frac{1}{x^2+x+2x+2}+...+\frac{1}{x^2+99x+100x+9900}=\frac{25}{51}\)
\(\Leftrightarrow\frac{1}{x\left(x+1\right)}+\frac{1}{x\left(x+1\right)+2\left(x+1\right)}+....+\frac{1}{x\left(x+99\right)+100\left(x+99\right)}=\frac{25}{51}\)
\(\Leftrightarrow\frac{1}{x\left(x+1\right)}+\frac{1}{\left(x+1\right)\left(x+2\right)}+...+\frac{1}{\left(x+99\right)\left(x+100\right)}=\frac{25}{21}\)
\(\Leftrightarrow\frac{1}{x}-\frac{1}{x+1}+\frac{1}{x+1}-\frac{1}{x+2}+...+\frac{1}{x+99}-\frac{1}{x+100}=\frac{25}{21}\)
\(\Leftrightarrow\frac{1}{x}-\frac{1}{x+100}=\frac{25}{21}\)
\(\Leftrightarrow\frac{x+100-x}{x\left(x+100\right)}=\frac{25}{21}\)
\(\Leftrightarrow\frac{100}{x\left(x+100\right)}=\frac{25}{21}\)
\(\Leftrightarrow25x^2+2500x=2100\)
\(\Leftrightarrow x^2+100x-84=0\)
\(\Leftrightarrow x^2+2.x.50+50^2-50^2-84=0\)
\(\Leftrightarrow\left(x+50\right)^2-2584=0\)
\(\Leftrightarrow\left(x+50-2\sqrt{646}\right)\left(x+50+2\sqrt{646}\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}x=-50+2\sqrt{646}\\x=-50-2\sqrt{646}\end{cases}}\)
Vậy ...
ĐK: x \(\ne\)-1; x \(\ne\)2
\(\frac{x+2}{x+1}+\frac{3}{x-2}=\frac{3}{x^2-x-2}+1\)
<=> \(\frac{\left(x+2\right)\left(x-2\right)}{\left(x+1\right)\left(x-2\right)}+\frac{3\left(x+1\right)}{\left(x+1\right)\left(x-2\right)}=\frac{3}{\left(x+1\right)\left(x-2\right)}+\frac{\left(x+1\right)\left(x-2\right)}{\left(x+1\right)\left(x-2\right)}\)
<=> x2 - 4 + 3x + 3 = 3 + x2 - x - 2
<=> x2 + 3x - x2 + x = 1 + 1
<=> 4x = 2
<=> x = 1/2
Vậy S = {1/2}
0.5 nha bạn
Chúc bạn học tốt! :)
\(\frac{x+2}{x+1}+\frac{3}{x-2}=\frac{3}{x^2-x-x}+1\)
\(\Leftrightarrow\frac{\left(x+2\right)\left(x-2\right)}{\left(x+1\right)\left(x-2\right)}+\frac{3\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}=\frac{3}{\left(x+1\right)\left(x-2\right)}+\frac{\left(x+1\right)\left(x-2\right)}{\left(x+1\right)\left(x-2\right)}\)
\(\Rightarrow x^2-4+3x+3=3+x^2-2x+x-2\)
\(\Leftrightarrow x^2-x^2+3x+2x-x=1+4-3\)
\(\Leftrightarrow4x=2\)
\(\Leftrightarrow x=\frac{1}{2}\)