The problem asks us to solve the equation $\frac{1}{4x} - \frac{3}{x} = \frac{-11}{x+12}$ for $x$. We need to find the value of $x$ that satisfies the equation, or determine if there is no solution.

1. Problem Description

The problem asks us to solve the equation $\frac{1}{4x} - \frac{3}{x} = \frac{-11}{x+12}$ for $x$. We need to find the value of $x$ that satisfies the equation, or determine if there is no solution.

2. Solution Steps

First, we find a common denominator for the left side of the equation:

$\frac{1}{4x} - \frac{3}{x} = \frac{1}{4x} - \frac{12}{4x} = \frac{1-12}{4x} = \frac{-11}{4x}$

Now the equation is:

$\frac{-11}{4x} = \frac{-11}{x+12}$

If $-11 \neq 0$, we can divide both sides by $-11$:

$\frac{1}{4x} = \frac{1}{x+12}$

Cross-multiply:

$x+12 = 4x$

Subtract $x$ from both sides:

$12 = 3x$

Divide by 3:

$x = \frac{12}{3} = 4$

Now, we need to check if this value of $x$ makes any of the denominators zero.

$4x = 4(4) = 16 \neq 0$

$x = 4 \neq 0$

$x+12 = 4+12 = 16 \neq 0$

So, $x=4$ is a valid solution.

We plug in $x=4$ to verify the solution:

$\frac{1}{4(4)} - \frac{3}{4} = \frac{1}{16} - \frac{3}{4} = \frac{1}{16} - \frac{12}{16} = \frac{-11}{16}$

$\frac{-11}{4+12} = \frac{-11}{16}$

Since both sides are equal, $x=4$ is the solution.

3. Final Answer

A. x = 4