1:

ĐKXĐ: \(x\notin\left\{3;-2;1\right\}\)

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

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

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

a) đK: \(x\ne0;2\)

B = \(\dfrac{3x-4}{x\left(x-2\right)}.\dfrac{x\left(x-2\right)}{x^2-4-x^2}=\dfrac{3x-4}{-4}=\dfrac{4-3x}{4}\) \(\dfrac{x-4+2x}{x\left(x-2\right)}:\dfrac{\left(x-2\right)\left(x+2\right)-x^2}{x\left(x-2\right)}\)

= \(\dfrac{3x-4}{x\left(x-2\right)}.\dfrac{x\left(x-2\right)}{x^2-4-x^2}=\dfrac{4-3x}{4}\)

b) Thay x = -2 (TMDK) vào B, ta có:

\(B=\dfrac{4-3.\left(-2\right)}{4}=\dfrac{4+6}{4}=\dfrac{5}{2}\)

c) Để \(\left|B\right|-2x=5\)

<=> \(\left|\dfrac{4-3x}{4}\right|-2x=5\)

TH1: \(x\le\dfrac{4}{3}\)

<=> \(\left|\dfrac{4-3x}{4}\right|=\dfrac{4-3x}{4}\)

PT <=> \(\dfrac{4-3x}{4}-2x=5\)

<=> \(\dfrac{4-3x-8x}{4}=5\)

<=> \(4-11x=20\)

<=> x = \(\dfrac{-16}{11}\) (Tm)

TH2: \(x>\dfrac{4}{3}\)

<=> \(\left|\dfrac{4-3x}{4}\right|=\dfrac{3x-4}{4}\)

PT <=> \(\dfrac{3x-4}{4}-2x=5\)

<=> \(\dfrac{3x-4-8x}{4}=5\)

<=> \(-5x-4=20\)

<=> \(x=\dfrac{-24}{5}\left(l\right)\)

d) Xét (2-x)B = \(\dfrac{\left(2-x\right)\left(4-3x\right)}{4}\)  = \(\dfrac{3x^2-10x+8}{4}\)

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

Mà \(3\left(x-\dfrac{5}{3}\right)^2\ge\) 0

=> (2-x)B \(\ge\dfrac{\dfrac{-1}{3}}{4}=\dfrac{-1}{12}\)

Dấu "=" <=> x = \(\dfrac{5}{3}\left(tm\right)\)

e) Số nguyên âm lớn nhất là -1

Để B = -1

<=> \(\dfrac{4-3x}{4}=-1\)

<=> 4 - 3x = -4
<=> \(x=\dfrac{8}{3}\left(tm\right)\)

g) 

TH1: \(x\le\dfrac{4}{3}\)

<=> \(\left|\dfrac{4-3x}{4}\right|=\dfrac{4-3x}{4}\)

BDT <=> \(\dfrac{4-3x}{4}< 2x-4\)

<=> \(4-3x< 8x-16\)

<=> \(x>\dfrac{20}{11}\left(l\right)\)

TH2: \(x>\dfrac{4}{3}\)

<=> \(\left|\dfrac{4-3x}{4}\right|=\dfrac{3x-4}{4}\)

BDT <=> \(\dfrac{3x-4}{4}< 2x-4\)

<=> \(3x-4< 8x-16\)

<=> x > \(\dfrac{12}{5}\)

KHDK: \(x>\dfrac{12}{5}\)

\(a,A=\dfrac{2x\left(x-3\right)+8\left(x+3\right)-2x-12}{\left(x-3\right)\left(x+3\right)}\cdot\dfrac{x+3}{x^2+6}\\ A=\dfrac{2x^2-6x+8x+24-2x-12}{\left(x-3\right)}\cdot\dfrac{1}{x^2+6}\\ A=\dfrac{2x^2+12}{\left(x-3\right)\left(x^2+6\right)}=\dfrac{2\left(x^2+6\right)}{\left(x-3\right)\left(x^2+6\right)}=\dfrac{2}{x-3}\)

\(b,A=5\Leftrightarrow\dfrac{2}{x-3}=5\Leftrightarrow5x-15=2\Leftrightarrow x=\dfrac{17}{5}\)

`B17:`

`a)` Với `x \ne +-3` có:

`A=[x+15]/[x^2-9]+2/[x+3]`

`A=[x+15+2(x-3)]/[(x-3)(x+3)]`

`A=[x+15+2x-6]/[(x-3)(x+3)]`

`A=[3x+9]/[(x-3)(x+3)]=3/[x-3]`

`b)A=[-1]/2<=>3/[x-3]=-1/2<=>-x+3=6<=>x=-3` (ko t/m)

   `=>` Ko có gtr nào của `x` t/m

`c)A in ZZ<=>3/[x-3] in ZZ`

   `=>x-3 in Ư_3`

 Mà `Ư_3={+-1;+-3}`

`@x-3=1=>x=4`

`@x-3=-1=>x=2`

`@x-3=3=>x=6`

`@x-3=-3=>x=0`

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`B18:`

`a)M=1/3`             `ĐK: x  \ne +-4`

`<=>(4/[x-4]-4/[x+4]).[x^2+8x+16]/32=1/3`

`<=>[4(x+4)-4(x-4)]/[(x-4)(x+4)].[(x+4)^2]/32=1/3`

`<=>32/[x-4].[x+4]/32=1/3`

`<=>3x+12=x-4`

`<=>x=-8` (t/m)