Question

question
solve for all values of x.
$2x - \frac{7}{x + 6} = x$
answer attempt 1 out of 2
additional solution no solution
$x = $

Explanation:

Step1: Isolate the rational term

Subtract $x$ from both sides.
$2x - x - \frac{7}{x+6} = 0$
$x - \frac{7}{x+6} = 0$

Step2: Eliminate the denominator

Multiply all terms by $x+6$ (where $x
eq -6$).
$x(x+6) - 7 = 0$

Step3: Expand and simplify to quadratic

Expand the left-hand side and rearrange.
$x^2 + 6x - 7 = 0$

Step4: Factor the quadratic

Factor the quadratic expression.
$(x+7)(x-1) = 0$

Step5: Solve for x

Set each factor equal to 0 and solve.
$x+7=0 \implies x=-7$; $x-1=0 \implies x=1$

Step6: Verify solutions

Check that $x=-7$ and $x=1$ do not make the original denominator zero, and satisfy the original equation. Both are valid.

Answer:

$x=-7$ and $x=1$