Question

the following rational equation has denominators that contain variables. for this equation, a. write the value or keeping the restrictions in mind, solve the equation.\\(\frac{5x}{x + 1}=2-\frac{5}{x + 1}\\)\
a. what are the value or values of the variable that makes the denominators zero?\\(x = -1\\)\\(\text{(simplify your answer. use a comma to separate answers as needed.)}\\)\
b. solve the equation. select the correct choice below and, if necessary, fill in the answer box to complete your ch\
\\(\bigcirc\\) a. the solution set is ( ).\\(\text{(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:

Part b: Solving the Equation

Step 1: Eliminate the Denominator

Multiply both sides of the equation \(\frac{5x}{x + 1}=2-\frac{5}{x + 1}\) by \(x + 1\) (note \(x
eq - 1\) from part a) to get rid of the denominators.
\((x + 1)\times\frac{5x}{x + 1}=(x + 1)\times2-(x + 1)\times\frac{5}{x + 1}\)
Simplify each term: \(5x = 2(x + 1)-5\)

Step 2: Expand and Simplify

Expand the right - hand side: \(5x=2x + 2-5\)
Combine like terms on the right - hand side: \(5x=2x-3\)

Step 3: Solve for \(x\)

Subtract \(2x\) from both sides: \(5x-2x=2x - 3-2x\)
Simplify: \(3x=-3\)
Divide both sides by 3: \(x=\frac{-3}{3}=-1\)

But from part a, we know that \(x=-1\) makes the denominator zero, so \(x = - 1\) is not a valid solution. So the solution set is empty.

Answer:

C. The solution set is \(\varnothing\)