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Bài làm:
PT:
đkxđ: \(x\ne0;x\ne2\)
Ta có: \(\frac{x+2}{x-2}=\frac{2}{x^2-2x}+\frac{1}{x}\)
\(\Leftrightarrow\frac{x\left(x+2\right)}{x\left(x-2\right)}=\frac{2}{x\left(x-2\right)}+\frac{x-2}{x\left(x-2\right)}\)
\(\Rightarrow x^2+2x=2+x-2\)
\(\Leftrightarrow x^2+x=0\)
\(\Leftrightarrow x\left(x+1\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}x=0\left(vl\right)\\x+1=0\end{cases}}\Rightarrow x=-1\)
BPT:
Ta có: \(\frac{x+1}{2}-x\le\frac{1}{2}\)
\(\Leftrightarrow\frac{x+1}{2}-x-\frac{1}{2}\le0\)
\(\Leftrightarrow\frac{x+1-2x-1}{2}\le0\)
\(\Leftrightarrow\frac{-x}{2}\le0\)
\(\Rightarrow-x\le0\)
\(\Rightarrow x\ge0\)
a) \(ĐKXĐ:\hept{\begin{cases}x\ne0\\x\ne2\end{cases}}\)
\(\frac{x+2}{x-2}=\frac{2}{x^2-2x}+\frac{1}{x}\)
\(\Leftrightarrow\frac{2}{x\left(x-2\right)}+\frac{1}{x}-\frac{x+2}{x-2}=0\)
\(\Leftrightarrow\frac{2+x-2-x^2-2x}{x\left(x-2\right)}=0\)
\(\Leftrightarrow-x^2-x=0\)
\(\Leftrightarrow-x\left(x+1\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}x=0\\x+1=0\end{cases}\Leftrightarrow\orbr{\begin{cases}x=0\left(ktm\right)\\x=-1\left(tm\right)\end{cases}}}\)
Vậy \(S=\left\{-1\right\}\)
b) \(\frac{x+1}{2}-x\le\frac{1}{2}\)
\(\Leftrightarrow x+1-2x-1\le0\)
\(\Leftrightarrow-x\le0\)
\(\Leftrightarrow x\ge0\)
Vậy \(x\ge0\)
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Điều kiện: x khác 0
Đặt \(\frac{x^2+1}{x}=t\Rightarrow\frac{x}{x^2+1}=\frac{1}{t}\)
Khi đó: \(\frac{x^2+1}{x}+\frac{x}{x^2+1}=\frac{5}{2}\)
\(\Leftrightarrow t+\frac{1}{t}=\frac{5}{2}\)
\(\Leftrightarrow\frac{t^2+1}{t}=\frac{5}{2}\Rightarrow2t^2+2=5t\)
\(\Leftrightarrow2t^2-5t+2=0\Leftrightarrow\left(2t-1\right)\left(t-2\right)=0\Leftrightarrow\orbr{\begin{cases}t=\frac{1}{2}\\t=2\end{cases}}\)
Nếu \(t=\frac{1}{2}\Rightarrow\frac{x^2+1}{x}=\frac{1}{2}\Rightarrow2x^2+2=x\)
\(\Leftrightarrow2x^2-x+2=0\)
Mà \(2x^2-x+2=2\left(x-\frac{1}{4}\right)^2+\frac{15}{8}>0\forall x\)
Nên \(x\in\varnothing\)
Nếu \(t=2\Rightarrow\frac{x^2+1}{x}=2\Rightarrow x^2-2x+1=0\)
\(\Leftrightarrow\left(x-1\right)^2=0\Leftrightarrow x=1\)(thỏa mãn ĐKXĐ)
Tập nghiệm của pt: \(S=\left\{1\right\}\)
\(\)
Theo BĐT AM-GM,ta có: \(x^2+1\ge2\left|x\right|\ge2x\Rightarrow\frac{x^2+1}{x}\ge2\)
Đặt \(\frac{x^2+t}{x}=t\left(t\ge2\right)\).Bài toán trở thành:
\(t+\frac{1}{t}=\frac{5}{2}\Leftrightarrow\left(\frac{1}{t}+\frac{t}{4}\right)+\frac{3t}{4}=\frac{5}{2}\)
Áp dụng BĐT AM-GM: \(VT\ge1+\frac{3t}{4}\ge1+\frac{6}{4}=\frac{5}{2}\)
Mà \(VT=\frac{5}{2}\) .Dấu "=" xảy ra khi \(\frac{1}{t}=\frac{t}{4}\Leftrightarrow t=2\Leftrightarrow\frac{x^2+1}{x}=2\Leftrightarrow x^2+1=2x\Leftrightarrow x=1\)
Vậy tập hợp nghiệm của phương trình: \(S=\left\{1\right\}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(\text{GIẢI :}\)
ĐKXĐ : \(x\ne1,\text{ }x\ne-2\).
\(\frac{2}{x-1}+\frac{1}{x+2}=\frac{x^2-x}{x-1}+\left(\text{-}x\right)\)
\(\Leftrightarrow\frac{2}{x-1}+\frac{1}{x+2}=\frac{x\left(x-1\right)}{x-1}+\left(\text{-}x\right)\)
\(\Leftrightarrow\frac{2}{x-1}+\frac{1}{x+2}=x+\left(\text{-}x\right)\)
\(\Leftrightarrow\frac{2}{x-1}+\frac{1}{x+2}=0\)
\(\Leftrightarrow\frac{2\left(x+2\right)}{\left(x-1\right)\left(x+2\right)}+\frac{x-1}{\left(x-1\right)\left(x+2\right)}=0\)
\(\Rightarrow2\left(x+2\right)+\left(x-1\right)=0\)
\(\Leftrightarrow2x+4+x-1\)
\(\Leftrightarrow3x+3=0\)
\(\Leftrightarrow3x=\text{-3}\Leftrightarrow x=\text{-1}\)
Vậy tập nghiệm của phương trình đã cho là \(S=\left\{-1\right\}\).
\(\frac{2}{x-1}+\frac{1}{x+2}=\frac{x^2-x}{x-1}+\left(-x\right)\left(đk:x\ne1;-2\right)\)
\(\frac{2\left(x+2\right)}{\left(x-1\right)\left(x+2\right)}+\frac{\left(x-1\right)}{\left(x+2\right)\left(x-1\right)}=\frac{x\left(x-1\right)}{x-1}-x\)
\(< =>\frac{2x+4+x-1}{\left(x-1\right)\left(x+2\right)}=x-x=0\)
\(< =>2x+4+x-1=0\)
\(< =>3x=1-4=-3\)
\(< =>x=\frac{-3}{3}=-1\left(tmđk\right)\)
Vậy nghiệm của phương trình trên là \(\left\{-1\right\}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(ĐKXĐ:\hept{\begin{cases}x\ne-1\\x\ne2\end{cases}}\)
\(\frac{2}{x+1}+\frac{1}{2-x}=\frac{3x-11}{x^2-x-2}\)
\(\Leftrightarrow\frac{2}{x+1}-\frac{1}{x-2}-\frac{3x-11}{\left(x+1\right)\left(x-2\right)}=0\)
\(\Leftrightarrow\frac{2\left(x-2\right)-\left(x+1\right)-\left(3x-11\right)}{\left(x+1\right)\left(x-2\right)}=0\)
\(\Leftrightarrow2x-4-x-1-3x+11=0\)
\(\Leftrightarrow-2x+6=0\)
\(\Leftrightarrow x=3\)
Vậy tập nghiệm của phương trình là \(S=\left\{3\right\}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
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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}
ĐKXĐ: x khác 0
\(x+\frac{1}{x}=x^2+\frac{1}{x^2}\Leftrightarrow\frac{x^3+x}{x^2}=\frac{x^4+1}{x^2}\)
=>x3+x=x4+1
<=>x4-x3-x+1=0
<=>x3(x-1)-(x-1)=0
<=>(x-1)(x3-1)=0
<=>(x-1)2(x2+x+1)=0
Mà \(x^2+x+1=x^2+x+\frac{1}{4}+\frac{3}{4}=\left(x+\frac{1}{2}\right)^2+\frac{3}{4}>0\)
=>x-1=0 <=> x=1 (tmđk)
vậy pt có tập nghiệm là S={1}