Question

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

Answer 1

Given:

`(x)/(x+6)-(x+1)/(x+2)=(x+2)/(x^2+8x+12)`

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

`\begin{gathered} (x)/(x+6)-(x+1)/(x+2)=(x+2)/(x^2+8x+12) \\ (x(x+2)-(x+1)(x+6))/((x+6)(x+2))=(x+2)/(x^2+8x+12) \\ \\ Expanding\text{ the denominator;} \\ (x(x+2)-(x+1)(x+6))/(x^2+8x+12)=(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=(8)/(-6) \\ x=-(4)/(3) \end{gathered}`

Therefore, the solution to the rational equation is;

`x=-(4)/(3)`

OPTION A is the correct answer.