Solve for \( d \) \( |-6 d-8|=34 \)

To solve the equation \( |-6d - 8| = 34 \) for \( d \), we'll consider the definition of absolute value. The absolute value equation \( |A| = B \) implies that \( A = B \) or \( A = -B \), provided that \( B \) is non-negative. Let's apply this to the given equation: \[ |-6d - 8| = 34 \] This leads to two separate equations: 1. **First Case:** \( -6d - 8 = 34 \) 2. **Second Case:** \( -6d - 8 = -34 \) We'll solve each case separately. --- ### **Case 1: \( -6d - 8 = 34 \)** 1. **Add 8 to both sides:** \[ -6d = 34 + 8 \] \[ -6d = 42 \] 2. **Divide both sides by -6:** \[ d = \frac{42}{-6} \] \[ d = -7 \] --- ### **Case 2: \( -6d - 8 = -34 \)** 1. **Add 8 to both sides:** \[ -6d = -34 + 8 \] \[ -6d = -26 \] 2. **Divide both sides by -6:** \[ d = \frac{-26}{-6} \] \[ d = \frac{13}{3} \] \[ d \approx 4.333 \] --- ### **Solution Summary** The equation \( |-6d - 8| = 34 \) has two solutions: \[ d = -7 \quad \text{and} \quad d = \frac{13}{3} \ (\text{or approximately } 4.333) \]