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Đặt \(A=\left(\frac{x^2}{x^3-4x}+\frac{6}{6-3x}+\frac{1}{x+2}\right):\left(x-2+\frac{10-x^2}{x+2}\right)\)
ĐKXĐ : \(x\ne\pm2,x\ne0\)
Ta có : \(A=\left(\frac{x^2}{x^3-4x}+\frac{6}{6-3x}+\frac{1}{x+2}\right):\left(x-2+\frac{10-x^2}{x+2}\right)\)
\(=\left(\frac{x^2}{x\left(x^2-4\right)}+\frac{6}{3\left(2-x\right)}+\frac{1}{x+2}\right):\left(\frac{\left(x-2\right)\left(x+2\right)+10-x^2}{x+2}\right)\)
\(=\left(\frac{x\cdot3-6\cdot\left(x+2\right)+3\cdot\left(x-2\right)}{3\left(x-2\right)\left(x+2\right)}\right):\left(\frac{x^2-4+10-x^2}{x+2}\right)\)
\(=\frac{-18}{3\left(x-2\right)\left(x+2\right)}:\left(-\frac{6}{x+2}\right)\)
\(=\frac{-6}{\left(x-2\right)\left(x+2\right)}\cdot\frac{x+2}{\left(-6\right)}=\frac{1}{x-2}\)
Để \(A\) nhận giá trị nguyên
\(\Leftrightarrow\frac{1}{x-2}\inℤ\) \(\Leftrightarrow1⋮x-2\) \(\Leftrightarrow x-2\inƯ\left(1\right)\)
\(\Leftrightarrow x-2\in\left\{-1,1\right\}\)
\(\Leftrightarrow x\in\left\{1,3\right\}\) ( Thỏa mãn ĐKXĐ )
Vậy : \(x\in\left\{1,3\right\}\) thì A nhận giá trị nguyên.
d) \(A>0\Leftrightarrow\frac{-1}{x-2}>0\)
\(\Leftrightarrow x-2< 0\) ( vì \(-1< 0\))
\(\Leftrightarrow x< 2\)
\(A=\left(\frac{x}{x^2-4}+\frac{2}{2-x}+\frac{1}{x+2}\right):\left(x-2+\frac{10-x^2}{x+2}\right)\)
\(A=\)\(\left[\frac{x}{\left(x-2\right)\left(x+2\right)}-\frac{2\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}+\frac{x-2}{\left(x-2\right)\left(x+2\right)}\right]\)
\(:\left[\frac{\left(x-2\right)\left(x+2\right)}{x+2}+\frac{10-x^2}{x+2}\right]\)
\(A=\frac{x-2x-4+x-2}{\left(x-2\right)\left(x+2\right)}:\left[\frac{x^2-4+10-x^2}{x+2}\right]\)
\(A=\frac{-6}{\left(x-2\right)\left(x+2\right)}:\frac{6}{x+2}\)
\(A=\frac{-6}{\left(x-2\right)\left(x+2\right)}.\frac{x+2}{6}\)
\(A=\frac{-1}{x-2}\)
a) \(A=\left(\frac{x+1}{x-1}-\frac{x-1}{x+1}+\frac{x^2-5x}{x^2-1}\right)\cdot\frac{x-3}{x}\left(x\ne\pm1;x\ne0\right)\)
\(\Leftrightarrow A=\left[\frac{\left(x+1\right)^2}{\left(x-1\right)\left(x+1\right)}-\frac{\left(x-1\right)^2}{\left(x-1\right)\left(x+1\right)}+\frac{x^2-5x}{\left(x-1\right)\left(x+1\right)}\right]\cdot\frac{x-3}{x}\)
\(\Leftrightarrow A=\left(\frac{x^2+2x+1-x^2+2x-1+x^2-5x}{\left(x-1\right)\left(x+1\right)}\right)\cdot\frac{x-3}{x}\)
\(\Leftrightarrow A=\frac{x^2-x}{\left(x-1\right)\left(x+1\right)}\cdot\frac{x-3}{x}\)
\(\Leftrightarrow A=\frac{x\left(x-1\right)\left(x-3\right)}{\left(x-1\right)\left(x+1\right)x}=\frac{x-3}{x+1}\)
Vậy \(A=\frac{x-3}{x+1}\left(x\ne\pm1;x\ne0\right)\)
b) \(A=\frac{x-3}{x+1}\left(x\ne\pm1;x\ne0\right)\)
Để A nhận giá trị nguyên thì x-3 chia hết chi x+1
=> (x+1)-4 chia hết chi x+1
=> 4 chia hết cho x+1
x nguyên => x+1 nguyên => x+1 thuộc Ư (4)={-4;-2;-1;1;2;4}
Ta có bảng
| x+1 | -4 | -2 | -1 | 1 | 2 | 4 |
| x | -5 | -3 | -2 | 0 | 1 | 3 |
| ĐCĐK | tm | tm | tm | ktm | ktm | tm |
Vậy x={-5;-3;-2;3} thì A đạt giá trị nguyên
c) I3x-1I=5
\(\Rightarrow\orbr{\begin{cases}3x-1=5\\3x-1=-5\end{cases}\Leftrightarrow\orbr{\begin{cases}3x=6\\3x=-4\end{cases}\Leftrightarrow}\orbr{\begin{cases}x=2\\x=\frac{-4}{3}\end{cases}}}\)
Đên đây thay vào rồi tính nhé
a) \(ĐKXĐ:\hept{\begin{cases}x\ne\pm1\\x\ne0\end{cases}}\)
\(A=\left(\frac{x+1}{x-1}-\frac{x-1}{x+1}+\frac{x^2-5x}{x^2-1}\right)\cdot\frac{x-3}{x}\)
\(\Leftrightarrow A=\frac{\left(x+1\right)^2-\left(x-1\right)^2+x^2-5x}{\left(x-1\right)\left(x+1\right)}\cdot\frac{x-3}{x}\)
\(\Leftrightarrow A=\frac{x^2+2x+1-x^2+2x-1+x^2-5x}{\left(x-1\right)\left(x+1\right)}\cdot\frac{x-3}{x}\)
\(\Leftrightarrow A=\frac{\left(x^2-x\right)\left(x-3\right)}{x\left(x-1\right)\left(x+1\right)}\)
\(\Leftrightarrow A=\frac{x-3}{x+1}\)
b) Để \(A\inℤ\)
\(\Leftrightarrow x-3⋮x+1\)
\(\Leftrightarrow x+1-4⋮x+1\)
\(\Leftrightarrow4⋮x+1\)
\(\Leftrightarrow x+1\inƯ\left(4\right)=\left\{\pm1;\pm2;\pm4\right\}\)
\(\Leftrightarrow x\in\left\{0;-2;-3;1;3;-5\right\}\)
Mà \(x\ne0;x\ne1\)
\(\Leftrightarrow x\in\left\{-2;-3;3;-5\right\}\)
Vậy để \(A\inℤ\Leftrightarrow x\in\left\{-2;-3;3;-5\right\}\)
c) Khi \(\left|3x-1\right|=5\)
\(\Leftrightarrow\orbr{\begin{cases}3x-1=5\\3x-1=-5\end{cases}}\)
\(\Leftrightarrow\orbr{\begin{cases}3x=6\\3x=-4\end{cases}}\)
\(\Leftrightarrow\orbr{\begin{cases}x=2\\x=-\frac{4}{3}\end{cases}}\)
Vì khi x = 2 hoặc x = -4/3 thì x không thuộc tập hợp các giá trị làm cho A nguyên
Vậy khi |3x - 1| = 5 thì để cho A nguyên \(\Leftrightarrow x\in\varnothing\)
\(ĐKXĐ:\hept{\begin{cases}x\ne\pm2\\x\ne0\end{cases}}\)
a) \(P=\left(\frac{x^2}{x^3-4x}+\frac{6}{6-3x}+\frac{1}{x+2}\right):\left(x-2+\frac{10-x^2}{x+2}\right)\)
\(\Leftrightarrow P=\left(\frac{x^2}{x\left(x-2\right)\left(x+2\right)}-\frac{6}{3\left(x-2\right)}+\frac{1}{x+2}\right):\frac{x^2-4+10-x^2}{x-2}\)
\(\Leftrightarrow P=\frac{x^2-2x\left(x+2\right)+x\left(x-2\right)}{x\left(x-2\right)\left(x+2\right)}:\frac{6}{x-2}\)
\(\Leftrightarrow P=\frac{x^2-2x^2-4x+x^2-2x}{x\left(x-2\right)\left(x+2\right)}\cdot\frac{x-2}{6}\)
\(\Leftrightarrow P=\frac{-6x}{6x\left(x+2\right)}\)
\(\Leftrightarrow P=\frac{-1}{x+2}\)
b) Khi \(\left|x\right|=\frac{3}{4}\)
\(\Leftrightarrow\orbr{\begin{cases}x=\frac{3}{4}\\x=-\frac{3}{4}\end{cases}}\)
\(\Leftrightarrow\orbr{\begin{cases}P=-\frac{1}{\frac{3}{4}+2}=-\frac{4}{11}\\P=-\frac{1}{-\frac{3}{4}+2}=-\frac{4}{5}\end{cases}}\)
c) Để P = 7
\(\Leftrightarrow-\frac{1}{x+2}=7\)
\(\Leftrightarrow7\left(x+2\right)=-1\)
\(\Leftrightarrow7x+14=-1\)
\(\Leftrightarrow7x=-15\)
\(\Leftrightarrow x=-\frac{15}{7}\)
Vậy để \(P=7\Leftrightarrow x=-\frac{15}{7}\)
d) Để \(P\inℤ\)
\(\Leftrightarrow1⋮x+2\)
\(\Leftrightarrow x+2\inƯ\left(1\right)=\left\{\pm1\right\}\)
\(\Leftrightarrow x\in\left\{-3;-1\right\}\)
Vậy để \(P\inℤ\Leftrightarrow x\in\left\{-3;-1\right\}\)