Let's solve the given equation step-by-step and determine the type of the equation.
Given:
[tex]\[
\frac{3 x}{x+2} - \frac{6}{x-2} = \frac{3 x^2 + 12}{x^2 - 4}
\][/tex]
Step 1: Simplify both sides of the equation.
First, simplify the left-hand side:
[tex]\[
\frac{3 x}{x+2} - \frac{6}{x-2}
\][/tex]
We can combine these by expressing them over a common denominator, which is [tex]\((x + 2)(x - 2)\)[/tex]:
[tex]\[
\frac{3 x (x - 2) - 6 (x + 2)}{(x + 2)(x - 2)}
\][/tex]
[tex]\[
= \frac{3 x^2 - 6x - 6x - 12}{x^2 - 4}
\][/tex]
[tex]\[
= \frac{3 x^2 - 12x - 12}{x^2 - 4}
\][/tex]
[tex]\[
= \frac{3 (x^2 - 4x - 4)}{x^2 - 4}
\][/tex]
Now, rewrite the right-hand side:
[tex]\[
\frac{3 x^2 + 12}{x^2 - 4}
\][/tex]
[tex]\[
= \frac{3 (x^2 + 4)}{x^2 - 4}
\][/tex]
Step 2: Equate the simplified forms.
We now have:
[tex]\[
\frac{3 (x^2 - 4x - 4)}{x^2 - 4} = \frac{3 (x^2 + 4)}{x^2 - 4}
\][/tex]
Since the denominators are the same, we equate the numerators:
[tex]\[
3 (x^2 - 4x - 4) = 3 (x^2 + 4)
\][/tex]
Step 3: Solve for [tex]\(x\)[/tex].
Divide both sides by 3:
[tex]\[
x^2 - 4x - 4 = x^2 + 4
\][/tex]
Subtract [tex]\(x^2\)[/tex] from both sides:
[tex]\[
-4x - 4 = 4
\][/tex]
Add 4 to both sides:
[tex]\[
-4x = 8
\][/tex]
Divide by -4:
[tex]\[
x = -2
\][/tex]
Step 4: Verify the solution.
Substitute [tex]\(x = -2\)[/tex] back into the original equation to check if it satisfies the equation:
[tex]\[
\frac{3 (-2)}{-2 + 2} - \frac{6}{-2 - 2} = \frac{3 (-2)^2 + 12}{(-2)^2 - 4}
\][/tex]
Evaluate the left-hand side:
[tex]\[
\frac{3 (-2)}{0} - \frac{6}{-4} \quad (\text{undefined due to division by zero})
\][/tex]
Since [tex]\(x = -2\)[/tex] results in an undefined expression in the original equation, it is not a valid solution.
Step 5: Determine the type of equation.
Since there are no valid solutions, the solution set is empty.
Hence, the solution set is:
[tex]\[
\varnothing
\][/tex]
And the type of the equation is:
[tex]\[
\text{Inconsistent equation}
\][/tex]
The correct choices are:
C. [tex]\(\varnothing\)[/tex]
Inconsistent equation
