a)  \(ĐKXĐ:x\ne\pm2\)

\(D=\frac{3x}{x-2}+\frac{2}{x+2}-\frac{14x-4}{x^2-4}:\frac{x\left(x-1\right)}{x+2}\)

\(\Leftrightarrow D=\frac{3x^2+6x+2x-4-14x+4}{\left(x-2\right)\left(x+2\right)}\cdot\frac{x+2}{x\left(x-1\right)}\)

\(\Leftrightarrow D=\frac{3x^2-6x}{x\left(x-1\right)\left(x-2\right)}\)

\(\Leftrightarrow D=\frac{3x\left(x-2\right)}{x\left(x-1\right)\left(x-2\right)}\)

\(\Leftrightarrow D=\frac{3}{x-1}\)

b) Khi \(\left|x-1\right|-3=0\)

\(\Leftrightarrow\left|x-1\right|=3\)

\(\Leftrightarrow\orbr{\begin{cases}x-1=3\\1-x=3\end{cases}}\)

\(\Leftrightarrow\orbr{\begin{cases}x=4\left(tm\right)\\x=-2\left(ktm\right)\end{cases}}\)

Thay \(x=4\)vào D ta được :\(D=\frac{3}{4-1}=1\)

c) Để D có giá trị nguyên

\(\Leftrightarrow\frac{3}{x-1}\)có giá trị nguyên

\(\Leftrightarrow x-1\inƯ\left(3\right)=\left\{\pm1;\pm3\right\}\)

\(\Leftrightarrow x\in\left\{0;2;-2;4\right\}\)

Loại bỏ giá trị \(x=\pm2\)không làm cho biểu thức có nghĩa

Vậy để D có giá trị nguyên \(\Leftrightarrow x\in\left\{0;4\right\}\)

Khi làm bài thì chỉnh lại giúp bạn cái đề: 

\(D=\left(\frac{3X}{X-2}+\frac{2}{X+2}-\frac{14X-4}{X^2-4}\right):\frac{X\left(X-1\right)}{X+2}\)

\(A=x^2+3x-5=x^2+3x+\frac{9}{4}-\frac{29}{4}\)

\(=\left(x+\frac{3}{2}\right)^2-\frac{29}{4}\ge-\frac{29}{4}\)

Vậy \(A_{min}=-\frac{29}{4}\Leftrightarrow x+\frac{3}{2}=0\Leftrightarrow x=-\frac{3}{2}\)

a, A xác định

\(\Leftrightarrow3x^3-19x^2+33x-9\ne0\)

\(\Leftrightarrow3x^3-x^2-18x^2+6x+27x-9\ne0\)

\(\Leftrightarrow x^2\left(3x-1\right)-6x\left(3x-1\right)+9\left(3x-1\right)\ne0\)

\(\Leftrightarrow\left(3x-1\right)\left(x-3\right)^2\ne0\Leftrightarrow\hept{\begin{cases}x\ne\frac{1}{3}\\x\ne3\end{cases}}\)

b, \(\frac{3x^3-14x^2+3x+36}{3x^2-19x^2+33x-9}=\frac{3x^2\left(x-3\right)-5x\left(x-3\right)-12\left(x-3\right)}{\left(3x-1\right)\left(x-3\right)^2}\)

\(=\frac{\left(3x^2-5x-12\right)\left(x-3\right)}{\left(3x-1\right)\left(x-3\right)^2}=\frac{\left(3x+4\right)\left(x-3\right)^2}{\left(3x-1\right)\left(x-3\right)^2}=\frac{3x+4}{3x-1}\)

\(A=0\Leftrightarrow\frac{3x+4}{3x-1}=0\Leftrightarrow3x+4=0\Leftrightarrow x=-\frac{4}{3}\) (thỏa mãn ĐKXĐ)

c, \(A=\frac{3x+4}{3x-1}=1+\frac{5}{3x-1}\in Z\Rightarrow5⋮\left(3x-1\right)\)

\(\Rightarrow3x-1\inƯ\left(5\right)=\left\{-5;-1;1;5\right\}\)

\(\Rightarrow x\in\left\{-\frac{4}{3};0;\frac{2}{3};2\right\}\)

Mà \(x\in Z,x\ne\left\{\frac{1}{3};3\right\}\Rightarrow x\in\left\{0;2\right\}\)

Bài của Hùng rất thông minh

Đang định có cách khác mà dài hơn cách Hùng nên thui

^^ 2k5 kết bạn nhé 

d, \(\frac{3x}{x+2}=\frac{3\left(x+2\right)-6}{x+2}=3-\frac{6}{x+2}\)

\(\Rightarrow x+2\inƯ\left(6\right)=\left\{\pm1;\pm2;\pm3;\pm6\right\}\)

e, \(C=\frac{A}{B}>0\Rightarrow\frac{3x}{x+2}.\frac{x+2}{x^2+2}=\frac{3x}{x^2+2}>0\)

\(\Rightarrow3x>0\Rightarrow x>0\)vì \(x^2+2>0\)

Kết hợp với đk vậy \(x>0;x\ne\pm2\)

f, vừa hỏi thầy, nên quay lại làm nốt :> 

f, Để \(\left|C\right|>C\Rightarrow C< 0\)vì \(\left|C\right|\ge0\)

\(\Rightarrow C=\frac{3x}{x^2+2}< 0\Rightarrow3x< 0\Leftrightarrow x< 0\)

Lời giải:

a.

\(A=\frac{(x\sqrt{x}-4x)-(\sqrt{x}-4)}{2(\sqrt{x}-4)(\sqrt{x}-2)(\sqrt{x}-1)}\)

ĐKXĐ: \(\left\{\begin{matrix} x\geq 0\\ \sqrt{x}-4\neq 0\\ \sqrt{x}-2\neq 0\\ \sqrt{x}-1\neq 0\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} x\geq 0\\ x\neq 16\\ x\neq 4\\ x\neq 1\end{matrix}\right.\)

\(A=\frac{x(\sqrt{x}-4)-(\sqrt{x}-4)}{2(\sqrt{x}-4)(\sqrt{2}-2)(\sqrt{x}-1)}=\frac{(x-1)(\sqrt{x}-4)}{2(\sqrt{x}-4)(\sqrt{x}-2)(\sqrt{x}-1)}\)

\(=\frac{(\sqrt{x}-1)(\sqrt{x}+1)(\sqrt{x}-4)}{2(\sqrt{x}-4)(\sqrt{x}-2)(\sqrt{x}-1)}=\frac{\sqrt{x}+1}{2(\sqrt{x}-2)}\)

b.

Với $x$ nguyên, để $A\in\mathbb{Z}$ thì $\sqrt{x}+1\vdots 2(\sqrt{x}-2)}$

$\Rightarrow \sqrt{x}+1\vdots \sqrt{x}-2$
$\Leftrightarrow \sqrt{x}-2+3\vdots \sqrt{x}-2$

$\Leftrightarrow 3\vdots \sqrt{x}-2$

$\Rightarrow \sqrt{x}-2\in\left\{\pm 1;\pm 3\right\}$

$\Rightarrow x\in\left\{1;9;25\right\}$

Thử lại thấy đều thỏa mãn.

a: \(A=\dfrac{x\left(\sqrt{x}-4\right)-\left(\sqrt{x}-4\right)}{2x\sqrt{x}-8x-6x+24\sqrt{x}+4\sqrt{x}-16}\)

\(=\dfrac{\left(\sqrt{x}-4\right)\left(x-1\right)}{\left(\sqrt{x}-4\right)\left(2x-6\sqrt{x}+4\right)}=\dfrac{x-1}{2x-6\sqrt{x}+4}\)

\(=\dfrac{x-1}{2\left(\sqrt{x}-1\right)\left(\sqrt{x}-2\right)}=\dfrac{\sqrt{x}+1}{2\sqrt{x}-4}\)

b: Để A nguyên thì \(2\sqrt{x}+2⋮2\sqrt{x}-4\)

\(\Leftrightarrow2\sqrt{x}-4\in\left\{2;-2;6\right\}\)

hay \(x\in\left\{9;1;25\right\}\)

a) \(A=\left(\dfrac{x}{x+3}-\dfrac{2}{x-3}+\dfrac{x^2-1}{9-x^2}\right):\left(2-\dfrac{x+5}{x+3}\right)\) (ĐK: \(x\ne\pm3\))

\(A=\left[\dfrac{x\left(x-3\right)}{\left(x+3\right)\left(x-3\right)}-\dfrac{2\left(x+3\right)}{\left(x+3\right)\left(x-3\right)}-\dfrac{x^2-1}{\left(x+3\right)\left(x-3\right)}\right]:\left(2+\dfrac{x+5}{x+3}\right)\)

\(A=\dfrac{x^2-3x-2x-6-x^2+1}{\left(x+3\right)\left(x-3\right)}:\dfrac{2\left(x+3\right)-\left(x+5\right)}{x+3}\)

\(A=\dfrac{-5x-5}{\left(x+3\right)\left(x-3\right)}\cdot\dfrac{x+3}{x+1}\)

\(A=\dfrac{-5\left(x+1\right)\left(x+3\right)}{\left(x+3\right)\left(x-3\right)\left(x+1\right)}\)

\(A=\dfrac{-5}{x-3}\)

b) Ta có: \(\left|x\right|=1\)

TH1: \(\left|x\right|=-x\) với \(x< 0\)

Pt trở thành:

\(-x=1\) (ĐK: \(x< 0\)) 

\(\Leftrightarrow x=-1\left(tm\right)\)

Thay \(x=-1\) vào A ta có:

\(A=\dfrac{-5}{x-3}=\dfrac{-5}{-1-3}=\dfrac{5}{4}\)

TH2: \(\left|x\right|=x\) với \(x\ge0\)

Pt trở thành:

\(x=1\left(tm\right)\) (ĐK: \(x\ge0\)) 

Thay \(x=1\) vào A ta có:

\(A=\dfrac{-5}{x-3}=\dfrac{-5}{1-2}=\dfrac{5}{2}\)

c) \(A=\dfrac{1}{2}\) khi:

\(\dfrac{-5}{x-3}=\dfrac{1}{2}\)

\(\Leftrightarrow-10=x-3\)

\(\Leftrightarrow x=-10+3\)

\(\Leftrightarrow x=-7\left(tm\right)\)

d) \(A\) nguyên khi:

\(\dfrac{-5}{x-3}\) nguyên

\(\Rightarrow x-3\inƯ\left(-5\right)\)

\(\Rightarrow x\in\left\{8;-2;2;4\right\}\)

a: \(A=\left(\dfrac{x}{x+3}-\dfrac{2}{x-3}+\dfrac{x^2-1}{9-x^2}\right):\left(2-\dfrac{x+5}{x+3}\right)\)

\(=\dfrac{x\left(x-3\right)-2\left(x+3\right)-x^2+1}{\left(x-3\right)\left(x+3\right)}:\dfrac{2x+6-x-5}{x+3}\)

\(=\dfrac{x^2-3x-2x-6-x^2+1}{\left(x-3\right)\left(x+3\right)}\cdot\dfrac{x+3}{x+1}\)

\(=\dfrac{-5x-5}{\left(x-3\right)}\cdot\dfrac{1}{x+1}=\dfrac{-5}{x-3}\)

b: |x|=1

=>x=-1(loại) hoặc x=1(nhận)

Khi x=1 thì \(A=\dfrac{-5}{1-3}=-\dfrac{5}{-2}=\dfrac{5}{2}\)

c: A=1/2

=>x-3=-10

=>x=-7

d: A nguyên

=>-5 chia hết cho x-3

=>x-3 thuộc {1;-1;5;-5}

=>x thuộc {4;2;8;-2}