a.
ĐKXĐ của A: \(x\ne\left\{-3;3\right\}\)
ĐKXĐ của B: \(x\ne-1\)
b.
\(A=\dfrac{2x\left(x-3\right)}{\left(x+3\right)\left(x-3\right)}+\dfrac{\left(x+1\right)\left(x+3\right)}{\left(x+3\right)\left(x-3\right)}+\dfrac{11x-3}{\left(x+3\right)\left(x-3\right)}\)
\(=\dfrac{2x\left(x-3\right)+\left(x+1\right)\left(x+3\right)+11x-3}{\left(x+3\right)\left(x-3\right)}\)
\(=\dfrac{3x^2+9x}{\left(x+3\right)\left(x-3\right)}=\dfrac{3x\left(x+3\right)}{\left(x+3\right)\left(x-3\right)}\)
\(=\dfrac{3x}{x-3}\)
c.
\(B< 1\Rightarrow\dfrac{x-3}{x+1}< 1\Rightarrow\dfrac{x-3}{x+1}-1< 0\)
\(\Rightarrow\dfrac{-4}{x+1}< 0\Rightarrow x+1>0\)
\(\Rightarrow x>-1\)
d.
\(P=AB=\dfrac{3x}{x-3}.\dfrac{x-3}{x+1}=\dfrac{3x}{x+1}\)
\(P=\dfrac{9}{2}\Rightarrow\dfrac{3x}{x+1}=\dfrac{9}{2}\Rightarrow2x=3x+3\)
\(\Rightarrow x=-3\)
e.
\(P=\dfrac{3x}{x+1}=\dfrac{3\left(x+1\right)-3}{x+1}=3-\dfrac{3}{x+1}\)
Để P nguyên \(\Rightarrow\dfrac{3}{x+1}\) nguyên \(\Rightarrow x+1=Ư\left(3\right)\)
\(\Rightarrow x+1=\left\{-3;-1;1;3\right\}\)
\(\Rightarrow x=\left\{-4;-2;0;2\right\}\)
