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{9}{5x + 25}=\frac{6}{x + 5}-\frac{3}{5}\\)
a. what is/are the value or values of the variable that make(s) the denominators zero?
x = - 5
(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.
○ a. the solution set is { }
(type an integer or a simplified fraction. use a comma to separate answers as needed.)
○ b. the solution set is {x| x is a real number}
○ c. the solution set is ∅

Explanation:

Step1: Factor the denominator

Factor $5x + 25$ to $5(x + 5)$. The equation becomes $\frac{9}{5(x + 5)}=\frac{6}{x + 5}-\frac{3}{5}$.

Step2: Find a common - denominator

The common denominator of the right - hand side is $5(x + 5)$. Rewrite the right - hand side: $\frac{6}{x + 5}\times\frac{5}{5}-\frac{3}{5}\times\frac{x + 5}{x + 5}=\frac{30}{5(x + 5)}-\frac{3(x + 5)}{5(x + 5)}$.

Step3: Set up the equation

The original equation $\frac{9}{5(x + 5)}=\frac{30}{5(x + 5)}-\frac{3(x + 5)}{5(x + 5)}$. Multiply both sides by $5(x + 5)$ (since $x
eq - 5$) to get $9 = 30-3(x + 5)$.

Step4: Expand and simplify

Expand the right - hand side: $9 = 30-(3x+15)=30 - 3x-15$. Combine like terms: $9 = 15-3x$.

Step5: Solve for x

Add $3x$ to both sides: $3x + 9=15$. Subtract 9 from both sides: $3x=15 - 9=6$. Divide both sides by 3: $x = 2$.

Answer:

A. The solution set is $\{2\}$