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}{x - 4}+5=\frac{27}{x - 4}\\)\\(x = 4\\)\\((\text{simplify your answer. use a comma to separate answers as needed.})\\)\\(\text{b. solve the equation. select the correct choice below and, if necessary, fill in the answer box to complete your choice.}\\)\\(\text{\\(\bigcirc\\) a. the solution set is \\(\\{\\ \\}\\).}\\)\\((\text{type an integer or a simplified fraction. use a comma to separate answers as needed.})\\)\\(\text{\\(\bigcirc\\) b. the solution set is \\(\\{x|x\text{ is a real number}\\}\\).}\\)\\(\text{\\(\bigcirc\\) c. the solution set is }\varnothing\text{.}\\)

Explanation:

Step1: Find restriction on x

Set denominator to zero: $x-4=0$
Solve for x: $x=4$

Step2: Eliminate denominators

Multiply all terms by $x-4$ (where $x
eq4$):
$$2 + 5(x-4) = 27$$

Step3: Simplify the equation

Expand and combine like terms:
$$2 + 5x - 20 = 27$$
$$5x - 18 = 27$$

Step4: Solve for x

Isolate x:
$$5x = 27 + 18$$
$$5x = 45$$
$$x = 9$$

Step5: Check validity

Verify $x=9$ does not make denominator zero: $9-4=5
eq0$, so it is valid.

Answer:

a. $x=4$
b. A. The solution set is $\{9\}$