Question

solve for x.
\\(\frac{x + 6}{x + 5}-1 = \frac{x - 1}{x - 5}\\)
if there is more than one solution, separate them with commas.
if there is no solution, click on
o solution\.
\\(x = \square\\)

Explanation:

Step1: Simplify left-hand side

Combine terms on LHS.
$$\frac{x+6}{x+5} - 1 = \frac{x+6 - (x+5)}{x+5} = \frac{1}{x+5}$$

Step2: Set up simplified equation

Equate simplified LHS to RHS.
$$\frac{1}{x+5} = \frac{x-1}{x-5}$$

Step3: Cross-multiply to eliminate denominators

Multiply both sides by $(x+5)(x-5)$.
$$1 \cdot (x-5) = (x-1)(x+5)$$

Step4: Expand both sides

Expand right-hand side polynomial.
$$x - 5 = x^2 + 5x - x - 5$$
$$x - 5 = x^2 + 4x - 5$$

Step5: Rearrange to standard quadratic form

Bring all terms to one side.
$$0 = x^2 + 4x - 5 - x + 5$$
$$x^2 + 3x = 0$$

Step6: Factor and solve for x

Factor out common term, find roots.
$$x(x + 3) = 0$$
$x=0$ or $x=-3$

Step7: Verify solutions

Check for extraneous roots (denominators ≠0).
For $x=0$: $0+5≠0$, $0-5≠0$ (valid)
For $x=-3$: $-3+5≠0$, $-3-5≠0$ (valid)

Answer:

$0, -3$