Question

question
solve for all values of x.
x - \frac{2x}{x + 6}=\frac{5}{x + 6}
answer
attempt 1 out of 2
additional solution no solution
x =

Explanation:

Step1: Multiply by common denominator

Multiply each term by $x + 6$:
$x(x + 6)-2x=5$

Step2: Expand and simplify

Expand $x(x + 6)$ to get $x^{2}+6x$. Then the equation becomes $x^{2}+6x-2x = 5$, which simplifies to $x^{2}+4x - 5=0$.

Step3: Factor the quadratic

Factor $x^{2}+4x - 5$ as $(x + 5)(x - 1)=0$.

Step4: Solve for x

Set each factor equal to zero:
If $x+5 = 0$, then $x=-5$; if $x - 1=0$, then $x = 1$.
We need to check for extraneous solutions. Since the original equation has a denominator of $x + 6$, when $x=-5$ and $x = 1$, the denominator $x+6
eq0$.

Answer:

$x=-5,1$