## Calculus with Applications (10th Edition)

$\dfrac{2}{x^{2}-2x-3}+\dfrac{5}{x^{2}-x-6}=\dfrac{1}{x^{2}+3x+2}$ Factor all three rational expressions completely: $\dfrac{2}{(x-3)(x+1)}+\dfrac{5}{(x-3)(x+2)}=\dfrac{1}{(x+2)(x+1)}$ Multiply the whole equation by $(x-3)(x+1)(x+2)$: $(x-3)(x+1)(x+2)\Big[\dfrac{2}{(x-3)(x+1)}+\dfrac{5}{(x-3)(x+2)}=\dfrac{1}{(x+2)(x+1)}\Big]$ $2(x+2)+5(x+1)=x-3$ $2x+4+5x+5=x-3$ Take all terms to the left side and simplify: $2x+4+5x+5-x+3=0$ $6x+12=0$ Take $12$ to the right side: $6x=-12$ Take $6$ to divide the right side: $x=-\dfrac{12}{6}$ $x=-2$ The original equation is undefined for $x=-2$, so this equation has no solution.