Question

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

Explanation:

Step1: Cross-multiply to eliminate denominators

$x(x-1) = -6(x+1)$

Step2: Expand both sides

$x^2 - x = -6x - 6$

Step3: Rearrange to standard quadratic form

$x^2 - x + 6x + 6 = 0$
$x^2 + 5x + 6 = 0$

Step4: Factor the quadratic

$(x+2)(x+3) = 0$

Step5: Solve for x

Set each factor equal to 0:
$x+2=0 \implies x=-2$
$x+3=0 \implies x=-3$

Step6: Verify no extraneous solutions

Substitute $x=-2$: $\frac{-2-1}{-2+1}=\frac{-3}{-1}=3$, $\frac{-6}{-2}=3$ (valid)
Substitute $x=-3$: $\frac{-3-1}{-3+1}=\frac{-4}{-2}=2$, $\frac{-6}{-3}=2$ (valid)

Answer:

$x=-2$ and $x=-3$