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

To solve the equation: \[ \frac{1}{4}x + 6 = -6 - \frac{4}{5}x \] **Step 1: Eliminate Fractions** Multiply both sides of the equation by the least common multiple (LCM) of the denominators 4 and 5, which is 20, to eliminate the fractions: \[ 20 \left( \frac{1}{4}x + 6 \right) = 20 \left( -6 - \frac{4}{5}x \right) \] This simplifies to: \[ 5x + 120 = -120 - 16x \] **Step 2: Collect Like Terms** Add \(16x\) to both sides to move all \(x\)-terms to the left: \[ 5x + 16x + 120 = -120 \] \[ 21x + 120 = -120 \] Next, subtract 120 from both sides to move the constants to the right: \[ 21x = -120 - 120 \] \[ 21x = -240 \] **Step 3: Solve for \(x\)** Divide both sides by 21: \[ x = \frac{-240}{21} \] Simplify the fraction by dividing the numerator and the denominator by 3: \[ x = \frac{-80}{7} \] **Final Answer:** \[ x = -\frac{80}{7} \] **Verification:** Plugging \(x = -\frac{80}{7}\) back into the original equation: \[ \frac{1}{4}\left(-\frac{80}{7}\right) + 6 = -6 - \frac{4}{5}\left(-\frac{80}{7}\right) \] Simplifying both sides: \[ -\frac{20}{7} + 6 = -6 + \frac{320}{35} \] \[ \frac{22}{7} = \frac{22}{7} \] Both sides are equal, confirming that \(x = -\frac{80}{7}\) is the correct solution. **Answer:** \[ x = -\frac{80}{7} \]