Solve the following equation. \[ \frac{1}{4} x+6=-6-\frac{4}{5} x \]

To solve the equation \( \frac{1}{4} x + 6 = -6 - \frac{4}{5} x \), we first want to isolate \( x \) on one side. 1. First, let's move the \( \frac{4}{5} x \) to the left side: \[ \frac{1}{4} x + \frac{4}{5} x + 6 = -6 \] 2. Next, we simplify the left side. To combine \( \frac{1}{4} x \) and \( \frac{4}{5} x \), we need a common denominator, which is 20: \[ \frac{1}{4} x = \frac{5}{20} x \quad \text{and} \quad \frac{4}{5} x = \frac{16}{20} x \] Thus, \[ \frac{5}{20} x + \frac{16}{20} x = \frac{21}{20} x \] We then rewrite our equation as: \[ \frac{21}{20} x + 6 = -6 \] 3. Next, isolate \( x \) by moving the 6 across: \[ \frac{21}{20} x = -6 - 6 \] This simplifies to: \[ \frac{21}{20} x = -12 \] 4. Now multiply both sides by the reciprocal of \( \frac{21}{20} \), which is \( \frac{20}{21} \): \[ x = -12 \times \frac{20}{21} \] 5. This simplifies to: \[ x = -\frac{240}{21} \] 6. Finally, we can reduce this fraction. The greatest common divisor of 240 and 21 is 3: \[ x = -\frac{80}{7} \] So, the solution to the equation is: \[ \boxed{-\frac{80}{7}} \]