Solve the following equation and show all the steps to solving. (x)/(x-2) + (x-1)/(x+1)=-1

Question

Solve the following equation and show all the steps to solving.


`(x)/(x-2) + (x-1)/(x+1)=-1`

Answer 1

`\bf \cfrac{x}{x-2}+\cfrac{x-1}{x+1}=-1\impliedby LCD\to (x-2)(x+1)\\\\ -------------------------------\\\\ \cfrac{x(x+1)+(x-1)(x-2)}{(x-2)(x+1)}=-1 \\\\\\ x^2+x+x^2-3x+2=-1(x-2)(x+1) \\\\\\ 2x^2-2x+2=-1(x^2-x-2)\implies 2x^2-2x+2=-x^2+x+2 \\\\\\ 3x^2-3x=0\implies 3x(x-1)=0\implies \begin{cases} 3x=0\implies &x=0\\ x-1=0\implies &x=1 \end{cases}`