Đáp án:
$a)B=\dfrac{8(x+2)}{(x-3)(x+7)}\\ b)B(-1)=-\dfrac{1}{3}\\ c)x=-3 \pm 2\sqrt{19}\\ d)\Leftrightarrow \left[\begin{array}{l} x>3 \\ -7<x<-2\end{array} \right.$
Giải thích các bước giải:
$B=\left(\dfrac{x^2+1}{x^2-9}-\dfrac{x}{x+3}+\dfrac{5}{x-3}\right):\left(\dfrac{2x+10}{x+3}-1\right)\\ ĐKXĐ:\left\{\begin{array}{l} x^2-9 \ne 0\\ x-3 \ne 0 \\ x+3 \ne 0 \\ \dfrac{2x+10}{x+3}-1 \ne 0\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} (x-3)(x+3) \ne 0\\ x-3 \ne 0 \\ x+3 \ne 0\\\dfrac{2x+10-(x+3)}{x+3} \ne 0 \end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} x \ne \pm 3 \\\dfrac{x+7}{x+3} \ne 0 \end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} x \ne \pm 3 \\ x \ne -7 \end{array} \right.\\ a)\left(\dfrac{x^2+1}{x^2-9}-\dfrac{x}{x+3}+\dfrac{5}{x-3}\right):\left(\dfrac{2x+10}{x+3}-1\right)\\ =\left(\dfrac{x^2+1}{(x-3)(x+3)}-\dfrac{x}{x+3}+\dfrac{5}{x-3}\right):\dfrac{2x+10-(x+3)}{x+3}\\ =\left(\dfrac{x^2+1}{(x-3)(x+3)}-\dfrac{x(x-3)}{x+3(x-3)}+\dfrac{5(x+3)}{(x-3)(x+3)}\right):\dfrac{2x+10-(x+3)}{x+3}\\ =\dfrac{x^2+1-x(x-3)+5(x+3)}{(x-3)(x+3)}:\dfrac{x+7}{x+3}\\ =\dfrac{8x+16}{(x-3)(x+3)}.\dfrac{x+3}{x+7}\\ =\dfrac{8(x+2)}{(x-3)(x+7)}\\ b)|x-1|=2\\ \Leftrightarrow \left[\begin{array}{cc} x-1=2 , x>1 \\ 1-x=2, x<1\end{array} \right.\\ \Leftrightarrow \left[\begin{array}{cc} x=3 \\ x=-1\end{array} \right.\\ x=3 \text{làm B không xác định nên không tính được } B(3)\\ B(-1)=\dfrac{8(-1+2)}{(-1-3)(-1+7)}=-\dfrac{1}{3}\\ c)B=\dfrac{x+5}{6}\\ \Leftrightarrow \dfrac{8(x+2)}{(x-3)(x+7)}=\dfrac{x+5}{6}\\ \Leftrightarrow \dfrac{8(x+2)}{(x-3)(x+7)}-\dfrac{x+5}{6}=0\\ \Leftrightarrow \dfrac{48(x+2)}{6(x-3)(x+7)}-\dfrac{(x+5)(x-3)(x+7)}{6(x-3)(x+7)}=0\\ \Leftrightarrow \dfrac{48(x+2)-(x+5)(x-3)(x+7)}{6(x-3)(x+7)}=0\\ \Leftrightarrow \dfrac{ - x^3- 9 x^2 + 49 x+201}{6(x-3)(x+7)}=0\\ \Leftrightarrow - x^3- 9 x^2 + 49 x+201=0\\ \Leftrightarrow - x^3-3x^2- 6 x^2 -18x+ 67 x+201=0\\ \Leftrightarrow - x^2(x+3)- 6 x(x+3)+ 67 (x+3)=0\\ \Leftrightarrow (x+3)(- x^2- 6 x+ 67)=0\\ \Leftrightarrow - x^2- 6 x+ 67=0(Do \ x \ne -3)\\ \Leftrightarrow x=-3 \pm 2\sqrt{19}\\ d)B>0\\ \Leftrightarrow \dfrac{8(x+2)}{(x-3)(x+7)}>0\\ \Leftrightarrow \dfrac{x+2}{(x-3)(x+7)}>0\\ \Leftrightarrow \left[\begin{array}{l} \left\{\begin{array}{l} x+2>0\\ x-3>0 \\ x+7>0\end{array} \right.\\ \left\{\begin{array}{l} x+2>0\\ x-3<0 \\ x+7<0\end{array} \right. \\ \left\{\begin{array}{l} x+2<0\\ x-3>0 \\ x+7<0\end{array} \right.\\\left\{\begin{array}{l} x+2<0\\ x-3<0 \\ x+7>0\end{array} \right.\end{array} \right.\\ \Leftrightarrow \left[\begin{array}{l} \left\{\begin{array}{l} x>-2\\ x>3 \\ x>-7\end{array} \right.\\ \left\{\begin{array}{l} x>-2\\ x<3 \\ x<-7\end{array} \right. \\ \left\{\begin{array}{l} x<-2\\ x>3\\ x<-7\end{array} \right.\\\left\{\begin{array}{l} x<-2\\ x<3 \\ x>-7\end{array} \right.\end{array} \right.\\ \Leftrightarrow \left[\begin{array}{l} x>3 \\ -7<x<-2\end{array} \right.$
