Question

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

Explanation:

Step1: Eliminate denominators

Multiply all terms by $x+1$ (where $x
eq -1$):
$$x(x+1) - (2x+1) = 1$$

Step2: Expand and simplify left side

$$x^2 + x - 2x - 1 = 1$$
$$x^2 - x - 1 = 1$$

Step3: Rearrange to standard quadratic form

Subtract 1 from both sides:
$$x^2 - x - 2 = 0$$

Step4: Factor the quadratic

$$(x-2)(x+1) = 0$$

Step5: Identify potential solutions

Set each factor equal to 0:
$x-2=0 \implies x=2$; $x+1=0 \implies x=-1$

Step6: Check for extraneous solutions

$x=-1$ makes original denominators 0, so it is invalid.

Answer:

$x=2$