Question

solve the equation. then determine whether the equation is an identity, a conditional equation, or an inconsistent equation.\\(\frac{4x}{x - 3}=\frac{12}{x - 3}+6\\)\\(\\)\\(\\)what is the solution? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\\(\\)\\(\\)\\(\\)a. the equation has a single solution. the solution set is \\(\\)\\(\\)b. the solution set is \\(\\{x|x\\) is a real number\\(\\}\\).\\(\\)\\(\\)c. the solution set is \\(\varnothing\\).

Explanation:

Step1: Eliminate denominators

Multiply both sides by $x-3$ (where $x
eq 3$):
$$4x = 12 + 6(x-3)$$

Step2: Expand right-hand side

Distribute the 6:
$$4x = 12 + 6x - 18$$

Step3: Simplify right-hand side

Combine constant terms:
$$4x = 6x - 6$$

Step4: Isolate x terms

Subtract $6x$ from both sides:
$$4x - 6x = -6$$
$$-2x = -6$$

Step5: Solve for x

Divide by $-2$:
$$x = \frac{-6}{-2} = 3$$

Step6: Check validity

$x=3$ makes the original denominator $x-3=0$, which is undefined. So there is no valid solution.

Answer:

C. The solution set is $\emptyset$

This is an inconsistent equation, as no real value of $x$ satisfies it.