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\\)\\(\\)\\(\\)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\\)\\(\\)determine whether the equation is an identity, a conditional equation, or an inconsistent equation.\\(\\)a. inconsistent equation\\(\\)b. identity\\(\\)c. conditional equation

Explanation:

Step1: Eliminate denominator

Multiply both sides by $x-3$ ($x
eq3$):
$$4x = 12 + 6(x-3)$$

Step2: Expand right-hand side

Simplify the parentheses:
$$4x = 12 + 6x - 18$$

Step3: Combine like terms

Simplify constants on right:
$$4x = 6x - 6$$

Step4: Isolate x terms

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

Step5: Solve for x

Divide by $-2$:
$$x = 3$$

Step6: Check domain validity

$x=3$ makes original denominator $x-3=0$, so it is not a valid solution.

Answer:

The solution set is $\emptyset$
A. Inconsistent equation