Question

$\frac{1}{(x+4)}-\frac{1}{(x-7)}=\frac{11}{30},(x\ne -4,7)$

Answer 1

$$\begin{gathered} \mathrm{Given}: \\ \frac{1}{(x+4)}-\frac{1}{(x-7)}=\frac{11}{30} \\ \Rightarrow \frac{(x-7)-(x+4)}{(x+4)(x-7)}=\frac{11}{30} \\ \Rightarrow \frac{-11}{(x^2-3x-28)}=\frac{11}{30} \\ \Rightarrow \frac{-1}{(x^2-3x-28)}=\frac{1}{30}\left[\mathrm{O}\text{n dividing both side by 11}\right] \\ \Rightarrow (x^2-3x-28)=-30[\mathrm{On}\text{cross multiplying}] \\ \Rightarrow x^2-3x+2=0 \\ \Rightarrow x^2-2x-x+2=0 \\ \Rightarrow x(x-2)-1(x-2)=0 \\ \Rightarrow (x-1)(x-2)=0 \\ \Rightarrow x-1=0\text{or}x-2=0 \\ \Rightarrow x=1\text{or}x=2 \\ \mathrm{H}\text{ence, the roots of the equation are}1\text{and}2. \end{gathered}$$