Solve the compound inequality \[ 3 x \geq 4 \text { or }-\frac{5}{8} x-5>6 \]

To solve the compound inequality \[ 3x \geq 4 \quad \text{or} \quad -\frac{5}{8}x - 5 > 6, \] we will solve each part separately. **Step 1: Solve the first inequality \(3x \geq 4\)** To isolate \(x\), divide both sides by 3: \[ x \geq \frac{4}{3}. \] **Step 2: Solve the second inequality \(-\frac{5}{8}x - 5 > 6\)** First, add 5 to both sides: \[ -\frac{5}{8}x > 6 + 5, \] \[ -\frac{5}{8}x > 11. \] Next, multiply both sides by \(-\frac{8}{5}\). Remember that multiplying or dividing by a negative number reverses the inequality: \[ x < 11 \cdot -\frac{8}{5}. \] Calculating the right side: \[ 11 \cdot -\frac{8}{5} = -\frac{88}{5}. \] Thus, we have: \[ x < -\frac{88}{5}. \] **Step 3: Combine the results** The solution to the compound inequality is: \[ x \geq \frac{4}{3} \quad \text{or} \quad x < -\frac{88}{5}. \] **Final Answer:** The solution set is \[ x < -\frac{88}{5} \quad \text{or} \quad x \geq \frac{4}{3}. \]