Bài 2:
\(\frac{3x-2}{x^2+1}>0\)
=>3x-2>0
=>3x>2
=>\(x>\frac23\)
Bài 3: \(\frac{x^2-x+1}{x+2}<0\)
mà \(x^2-x+1=x^2-x+\frac14+\frac34=\left(x-\frac12\right)^2+\frac34\ge\frac34>0\forall x\)
nên x+2<0
=>x<-2
Bài 4:
a: \(\frac{8x^3y^2}{6x^2\cdot\left(-y\right)^3}=\frac{-8x^3y^2}{6x^2y^3}=\frac{-2x^2y^2\cdot4x}{2x^2y^2\cdot3y}=\frac{-4x}{3y}\)
b: \(\frac{-7\cdot\left(-x\right)^4\cdot y}{14x^3y^2}=\frac{-7x^4y}{14x^3y^2}=\frac{-7x^3y\cdot x}{7x^3y\cdot2y}=\frac{-x}{2y}\)
c: \(\frac{-2x^2+2x}{x-1}=\frac{-2x\left(x-1\right)}{x-1}=-2x\)
d: \(\frac{\left(2x+3\right)^2-x^2}{x^2-1}=\frac{\left(2x+3-x\right)\left(2x+3+x\right)}{\left(x-1\right)\left(x+1\right)}\)
\(=\frac{\left(x+3\right)\left(3x+3\right)}{\left(x-1\right)\left(x+1\right)}=\frac{\left(x+3\right)\cdot3\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}=\frac{3\left(x+3\right)}{x-1}\)
e: \(\frac{x^3-3x^2-x+3}{x^2-3x}\)
\(=\frac{x^2\left(x-3\right)-\left(x-3\right)}{x\left(x-3\right)}=\frac{\left(x-3\right)\left(x^2-1\right)}{x\left(x-3\right)}=\frac{x^2-1}{x}\)
f: \(\frac{x^2-y^2}{x^2+2x-y^2-2y}\)
\(=\frac{\left(x-y\right)\left(x+y\right)}{\left(x^2-y^2\right)+2x-2y}=\frac{\left(x-y\right)\left(x+y\right)}{\left(x-y\right)\left(x+y\right)+2\left(x-y\right)}\)
\(=\frac{\left(x-y\right)\left(x+y\right)}{\left(x-y\right)\left(x+y+2\right)}=\frac{x+y}{x+y+2}\)
g: \(\frac{10xy^2\left(2x-1\right)^3}{12x^3y\left(1-2x\right)}=\frac{-10xy^2\left(2x-1\right)^3}{12x^3y\left(2x-1\right)}=\frac{-2xy\left(2x-1\right)\cdot5y\left(2x-1\right)^2}{2xy\left(2x-1\right)\cdot6x^2}=\frac{-5y\left(2x-1\right)^2}{6x^2}\)
h: \(\frac{x^2+7x+12}{x^2+5x+6}\)
\(=\frac{\left(x+3\right)\left(x+4\right)}{\left(x+3\right)\left(x+2\right)}=\frac{x+4}{x+2}\)
i: \(\frac{\left(3x+2\right)^2-\left(x+2\right)^2}{x^3-x^2}\)
\(=\frac{\left(3x+2+x+2\right)\left(3x+2-x-2\right)}{x^2\left(x-1\right)}=\frac{\left(4x+4\right)\cdot2x}{x^2\left(x-1\right)}\)
\(=\frac{2\left(4x+4\right)}{x\left(x-1\right)}=\frac{8x+8}{x^2-x}\)
m: \(\frac{20x^2-45}{\left(2x+3\right)^2}=\frac{5\left(4x^2-9\right)}{\left(2x+3\right)^2}=\frac{5\left(2x-3\right)\left(2x+3\right)}{\left(2x+3\right)^2}=\frac{5\left(2x-3\right)}{2x+3}\)
n: \(\frac{5x^2-10xy}{2\left(2y-x\right)^3}=\frac{5x\left(x-2y\right)}{-2\left(x-2y\right)^3}=\frac{-5x}{2\left(x-2y\right)^2}\)
o: \(\frac{6x^2-x-15}{25-9x^2}=\frac{6x^2-10x+9x-15}{-\left(3x-5\right)\left(3x+5\right)}\)
\(=\frac{2x\left(3x-5\right)+3\left(3x-5\right)}{-\left(3x-5\right)\left(3x+5\right)}=\frac{\left(3x-5\right)\left(2x+3\right)}{-\left(3x-5\right)\left(3x+5\right)}\)
\(=\frac{-2x-3}{3x+5}\)
