Question

question
solve for all values of x.
$x - \frac{3x + 5}{x + 9} = 1$
answer attempt 1 out of 2
additional solution no solution
$x = \square$

Explanation:

Step1: Eliminate the denominator

Multiply all terms by $x+9$ (where $x
eq -9$):
$$(x)(x+9) - (3x+5) = 1(x+9)$$

Step2: Expand all expressions

$$x^2 + 9x - 3x - 5 = x + 9$$

Step3: Simplify left side

Combine like terms:
$$x^2 + 6x - 5 = x + 9$$

Step4: Rearrange to standard quadratic form

Move all terms to left side:
$$x^2 + 6x - 5 - x - 9 = 0$$
$$x^2 + 5x - 14 = 0$$

Step5: Factor the quadratic

Find two factors of -14 that sum to 5:
$$(x+7)(x-2) = 0$$

Step6: Solve for x

Set each factor equal to 0:
$x+7=0 \implies x=-7$; $x-2=0 \implies x=2$

Step7: Verify no excluded values

Neither $x=-7$ nor $x=2$ make $x+9=0$, so both are valid.

Answer:

$x=-7$ and $x=2$