Question

Which of the following represents all of the solutions to the rational equation

Answer 1

Solution:

Given:

$\frac{x}{x+6}-\frac{x+1}{x+2}=\frac{x+2}{x^2+8x+12}$

Expressing the left-hand side of the equation as a single fraction;

$\begin{gathered} \frac{x}{x+6}-\frac{x+1}{x+2}=\frac{x+2}{x^2+8x+12} \\ \frac{x(x+2)-(x+1)(x+6)}{(x+6)(x+2)}=\frac{x+2}{x^2+8x+12} \\ \\ Expanding\text{ the denominator;} \\ \frac{x(x+2)-(x+1)(x+6)}{x^2+8x+12}=\frac{x+2}{x^{2}+8x+12} \\ Equating\text{ the numerators;} \\ x(x+2)-(x+1)(x+6)=x+2 \end{gathered}$

Expanding and simplifying further;

$\begin{gathered} x^2+2x-(x^2+6x+x+6)=x+2 \\ x^2+2x-(x^2+7x+6)=x+2 \\ x^2-x^2+2x-7x-6=x+2 \\ -5x-6=x+2 \\ -5x-x=2+6 \\ -6x=8 \\ x=\frac{8}{-6} \\ x=-\frac{4}{3} \end{gathered}$

Therefore, the solution to the rational equation is;

$x=-\frac{4}{3}$

OPTION A is the correct answer.