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Bài 1:
a) đkxđ: \(x\ne0;x\ne\pm1\)
\(D=\left(\frac{1}{1-x}+\frac{1}{1+x}\right)\div\left(\frac{1}{1-x}-\frac{1}{1+x}\right)+\frac{1}{x+1}\)
\(D=\left[\frac{1+x+1-x}{\left(1-x\right)\left(1+x\right)}\right]\div\left[\frac{1+x-1+x}{\left(1-x\right)\left(1+x\right)}\right]+\frac{1}{x+1}\)
\(D=\frac{2}{\left(1-x\right)\left(1+x\right)}\div\frac{2x}{\left(1-x\right)\left(1+x\right)}+\frac{1}{x+1}\)
\(B=\frac{1}{x}+\frac{1}{x+1}\)
\(B=\frac{2x+1}{x+1}\)
b) Ta có: \(x^2-x=0\Leftrightarrow x\left(x-1\right)=0\Leftrightarrow\orbr{\begin{cases}x=0\\x-1=0\end{cases}}\Rightarrow\orbr{\begin{cases}x=0\\x=1\end{cases}}\) đều ko thỏa mãn đkxđ
c) Khi \(D=\frac{3}{2}\)
\(\Leftrightarrow\frac{2x+1}{x+1}=\frac{3}{2}\)
\(\Leftrightarrow4x+2=3x+3\Rightarrow x=1\) không thỏa mãn đkxđ
Bài 2: (Sửa đề tí nếu sai ib t lm lại nhé:)
a) đkxđ: \(x\ne\pm1\)
\(E=\left(\frac{x+1}{x-1}-\frac{x-1}{x+1}\right)\div\left(\frac{1}{x+1}-\frac{x}{1-x}+\frac{2}{x^2-1}\right)\)
\(E=\frac{\left(x+1\right)^2-\left(x-1\right)^2}{\left(x-1\right)\left(x+1\right)}\div\frac{x-1+x\left(x+1\right)+2}{\left(x-1\right)\left(x+1\right)}\)
\(E=\frac{x^2+2x+1-x^2+2x-1}{x-1+x^2+x+2}\)
\(E=\frac{4x}{\left(x+1\right)^2}\)
b) Ta có: \(x^2-9=0\Rightarrow\orbr{\begin{cases}x=3\\x=-3\end{cases}}\)
+ Nếu: \(x=3\)
=> \(E=\frac{4.3}{\left(3+1\right)^2}=\frac{3}{4}\)
+ Nếu: \(x=-3\)
=> \(E=\frac{4.\left(-3\right)}{\left(-3+1\right)^2}=-3\)
c) Để \(E=-3\)
\(\Leftrightarrow\frac{4x}{\left(x+1\right)^2}=-3\)
\(\Leftrightarrow4x=-3x^2-6x-3\)
\(\Leftrightarrow3x^2+10x+3=0\)
\(\Leftrightarrow\left(x+3\right)\left(3x+1\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}x+3=0\\3x+1=0\end{cases}}\Leftrightarrow\orbr{\begin{cases}x=-3\\x=-\frac{1}{3}\end{cases}}\)
d) Để \(E< 0\)
\(\Leftrightarrow\frac{4x}{\left(x+1\right)^2}< 0\) , mà \(\left(x+1\right)^2>0\left(\forall x\right)\)
=> Để E < 0 => \(4x< 0\Rightarrow x< 0\)
Vậy x < 0 thì E < 0
e) Ta có: \(E-x-3=0\)
\(\Leftrightarrow\frac{4x}{\left(x+1\right)^2}=x+3\)
\(\Leftrightarrow4x=\left(x^2+2x+1\right)\left(x+3\right)\)
\(\Leftrightarrow x^3+5x^2+7x+3-4x=0\)
\(\Leftrightarrow x^3+5x^2+3x+3=0\)
Đến đây bấm máy tính thôi, nghiệm k đc đẹp cho lắm:
\(x=-4,4798...\) ; \(x=-0,2600...+0,7759...\) ; \(x=-0,2600...-0,7759...\)
\(E=\frac{x+\sqrt{x}}{x-2\sqrt{x}+1}:\left(\frac{\sqrt{x}+1}{\sqrt{x}}-\frac{1}{1-\sqrt{x}}+\frac{2-x}{x-\sqrt{x}}\right)\)
\(E=\frac{\sqrt{x}\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)^2}:\) \(\left[\frac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)}+\frac{\sqrt{x}}{\left(\sqrt{x}-1\right)\sqrt{x}}+\frac{2-x}{\sqrt{x}\left(\sqrt{x}-1\right)}\right]\)
\(E=\frac{\sqrt{x}\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)^2}:\)\(\left[\frac{x-1+\sqrt{x}+2-x}{\sqrt{x}\left(\sqrt{x}-1\right)}\right]\)
\(E=\frac{\sqrt{x}\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)^2}:\frac{\sqrt{x}+1}{\sqrt{x}\left(\sqrt{x}-1\right)}\)
\(E=\frac{\sqrt{x}\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)^2}.\frac{\sqrt{x}\left(\sqrt{x}-1\right)}{\sqrt{x}+1}\)
\(E=\frac{x}{\sqrt{x}-1}\)
b) \(E>1\Leftrightarrow\frac{x}{\sqrt{x}-1}>1\)
\(\Leftrightarrow\frac{x}{\sqrt{x}-1}-1>0\)
\(\Leftrightarrow\frac{x}{\sqrt{x}-1}-\frac{\sqrt{x}-1}{\sqrt{x}-1}>0\)
\(\Leftrightarrow\frac{x-\sqrt{x}+1}{\sqrt{x}-1}>0\)
\(\Leftrightarrow\frac{x-2\sqrt{x}+1+\sqrt{x}}{\sqrt{x}-1}>0\)
\(\Leftrightarrow\frac{\left(\sqrt{x}-1\right)^2+\sqrt{x}}{\sqrt{x}-1}>0\)
\(\Rightarrow\sqrt{x}-1>0\) vì tử của phân số luôn \(\ge0\forall x\ge0\)
\(\Rightarrow x>1\)
kết hợp với ĐKXĐ \(x\ge0\Rightarrow x>1\)
vậy \(x>1\) thì \(E>1\)
I) Đk: x > 0 và x \(\ne\)9
\(D=\left(\frac{x+3}{x-9}+\frac{1}{\sqrt{x}+3}\right):\frac{\sqrt{x}}{\sqrt{x}-3}\)
\(D=\frac{x+3+\sqrt{x}-3}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\cdot\frac{\sqrt{x}-3}{\sqrt{x}}\)
\(D=\frac{\sqrt{x}\left(\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}+3\right)}=\frac{\sqrt{x}+1}{\sqrt{x}+3}\)
=> \(\frac{1}{D}=\frac{\sqrt{x}+3}{\sqrt{x}+1}=\frac{\sqrt{x}+1+2}{\sqrt{x}+1}=1+\frac{2}{\sqrt{x}+1}\)
Để 1/D nguyên <=> \(\frac{2}{\sqrt{x}+1}\in Z\)
<=> \(2⋮\left(\sqrt{x}+1\right)\) <=> \(\sqrt{x}+1\inƯ\left(2\right)=\left\{1;-1;2;-2\right\}\)
Do \(x>0\) => \(\sqrt{x}+1>1\) => \(\sqrt{x}+1=2\)
<=> \(\sqrt{x}=1\) <=> x = 1 (tm)
\(E=\left(\frac{x+2}{x\sqrt{x}+1}-\frac{1}{\sqrt{x}+1}\right)\cdot\frac{4\sqrt{x}}{3}\)
\(E=\frac{x+2-x+\sqrt{x}-1}{\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}\cdot\frac{4\sqrt{x}}{3}\)
\(E=\frac{\sqrt{x}+1}{\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}\cdot\frac{4\sqrt{x}}{3}=\frac{4\sqrt{x}}{3\left(x-\sqrt{x}+1\right)}\)
b) Với x\(\ge\)0; ta có:
\(E=\frac{8}{9}\) <=> \(\frac{4\sqrt{x}}{3\left(x-\sqrt{x}+1\right)}=\frac{8}{9}\)
<=> \(3\sqrt{x}=2x-2\sqrt{x}+2\)
<=> \(2x-4\sqrt{x}-\sqrt{x}+2=0\)
<=> \(\left(2\sqrt{x}-1\right)\left(\sqrt{x}-2\right)=0\)
<=> \(\orbr{\begin{cases}x=\frac{1}{4}\left(tm\right)\\x=4\left(tm\right)\end{cases}}\)
e) Ta có: \(E=\frac{4\sqrt{x}}{3\left(x-\sqrt{x}+1\right)}\ge0\forall x\in R\) (vì \(x-\sqrt{x}+1=\left(\sqrt{x}-\frac{1}{2}\right)^2+\frac{3}{4}>0\forall x\))
Dấu "=" xảy ra<=> x = 0
Vậy MinE = 0 <=> x = 0
Lại có: \(\frac{1}{E}=\frac{3\left(x-\sqrt{x}+1\right)}{4\sqrt{x}}=\frac{3}{4}\left(\sqrt{x}-1+\frac{1}{\sqrt{x}}\right)\ge\frac{3}{4}\left(2\sqrt{\sqrt{x}\cdot\frac{1}{\sqrt{x}}}-1\right)\)(bđt cosi)
=> \(\frac{1}{E}\ge\frac{3}{2}.\left(2-1\right)=\frac{3}{2}\)=> \(E\le\frac{2}{3}\)
Dấu "=" xảy ra<=> \(\sqrt{x}=\frac{1}{\sqrt{x}}\) <=> x = 1
Vậy MaxE = 2/3 <=> x = 1