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{x - 3}{2x} + 1 = \frac{x + 2}{x}\\)

a. what is/are the value or values of the variable that make(s) the denominators zero?
x = 0
(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 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 restriction on x

Set denominators equal to 0:
$2x=0$ and $x=0$
Solve for $x$: $x=0$

Step2: Eliminate denominators

Multiply all terms by $2x$ (LCD):
$$2x \cdot \frac{x-3}{2x} + 2x \cdot 1 = 2x \cdot \frac{x+2}{x}$$
Simplify: $x-3 + 2x = 2(x+2)$

Step3: Simplify both sides

Combine like terms and expand:
$3x - 3 = 2x + 4$

Step4: Isolate the variable x

Subtract $2x$ and add 3 to both sides:
$3x - 2x = 4 + 3$
$x=7$

Step5: Verify against restriction

Check that $x=7
eq 0$, so it is valid.

Answer:

a. $x=0$
b. A. The solution set is $\{7\}$