Question

solve the equation.$\frac{2x + 3}{2} - \frac{3x}{x - 4} = x$select the correct choice below and, if necessary, fill in the answer box to complete your choice.a. the solution set is {}.(simplify your answer. use a comma to separate answers as needed.)b. the solution is the empty set.

Explanation:

Step1: Eliminate denominators

Multiply all terms by $2(x-4)$:
$$(2x+3)(x-4) - 2(3x) = 2x(x-4)$$

Step2: Expand all expressions

$$2x^2 -8x +3x -12 -6x = 2x^2 -8x$$

Step3: Simplify both sides

Combine like terms:
$$2x^2 -11x -12 = 2x^2 -8x$$

Step4: Isolate variable terms

Subtract $2x^2$ from both sides:
$$-11x -12 = -8x$$

Step5: Solve for $x$

Rearrange to collect $x$ terms:
$$-12 = 3x$$
$$x = -4$$

Step6: Verify the solution

Substitute $x=-4$ into original equation:
Left side: $\frac{2(-4)+3}{2} - \frac{3(-4)}{-4-4} = \frac{-5}{2} - \frac{-12}{-8} = \frac{-5}{2} - \frac{3}{2} = -4$
Right side: $-4$. The solution is valid.

Answer:

A. The solution set is $\{-4\}$.