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{4}{5x + 25} = \frac{8}{x + 5} - \frac{4}{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 \\(\varnothing\\).

Explanation:

Step1: Find restrictions (denominator=0)

First denominator: $5x+25=0 \implies 5(x+5)=0 \implies x=-5$
Second denominator: $x+5=0 \implies x=-5$
Third denominator: $5
eq 0$ for all $x$

Step2: Simplify the equation

Factor $5x+25=5(x+5)$. Multiply all terms by $5(x+5)$ (the least common denominator) to eliminate fractions:
$$4 = 8 \times 5 - 4(x+5)$$

Step3: Expand and simplify right-hand side

$$4 = 40 - 4x - 20$$
$$4 = 20 - 4x$$

Step4: Solve for $x$

Rearrange to isolate $x$:
$$4x = 20 - 4$$
$$4x = 16$$
$$x = 4$$

Step5: Check against restrictions

$x=4$ does not equal $-5$, so it is a valid solution.

Answer:

a. $x=-5$
b. A. The solution set is $\{4\}$