Question

the following rational equation has denominators that contain variables. for this equation, a. write the value or values of the variable that make a denominator zero. these are the restrictions on the variable. b. keeping the restrictions in mind, solve the equation.\\(\frac{2}{9x + 27} = \frac{6}{x + 3} - \frac{4}{9}\\)\
a. what is/are the value or values of the variable that make(s) the denominators zero?\
x = -3\
(simplify your answer. use a comma to separate answers as needed.)\
b. select the correct choice below and, if necessary, fill in the answer box to complete your choice\
\\(\bigcirc\\) a. the solution set is \\{\\}.\
(type an integer or a simplified fraction. use a comma to separate answers as needed)\
\\(\bigcirc\\) b. the solution set is \\{x | x is a real number\\}.\
\\(\bigcirc\\) c. the solution set is \\(\varnothing\\).

Explanation:

Step1: Find restrictions (denominator=0)

First, factor denominators: $9x+27=9(x+3)$. Set denominators to 0:
$9(x+3)=0 \implies x+3=0 \implies x=-3$; $x+3=0 \implies x=-3$; 9 is never 0.

Step2: Simplify the equation

Multiply all terms by LCD $9(x+3)$ to eliminate denominators:
$$2 = 6 \cdot 9 - 4(x+3)$$

Step3: Expand and simplify right side

$$2 = 54 - 4x - 12$$
$$2 = 42 - 4x$$

Step4: Isolate the variable term

$$4x = 42 - 2$$
$$4x = 40$$

Step5: Solve for x

$$x = \frac{40}{4} = 10$$

Step6: Check against restrictions

$x=10
eq -3$, so it is valid.

Answer:

a. $x=-3$
b. A. The solution set is $\{10\}$